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When writer Lewis Carroll took Alice on her journeys down the rabbit hole toWonderland and through the lookingglass, she had many fantasticencounters with the tea-sipping MadHatter, a hookah-smoking Caterpillar, theWhite Rabbit, the Cheshire Cat, the Redand White Queens, and Tweedledum andTweedledee. On the surface, Carroll’swritings seem to be delightful nonsenseand mere children’s entertainment. Manypeople are quite surprised to learn thatAlice’s Adventures in Wonderland is asmuch an exercise in logic as it is a fantasyand that Lewis Carroll was actuallyCharles Dodgson, an Oxfordmathematician. Dodgson’s many writingsinclude the whimsical The Game of Logicand the brilliant Symbolic Logic, inaddition to Alice’s Adventures inWonderland and Through the LookingGlass.

WHAT WE WILL DO In This Chapter

WE’LL EXPLORE DIFFERENT TYPES OF LOGIC ORREASONING:

• Deductive reasoning involves the application of ageneral statement to a specific case; this type oflogic is typified in the classic arguments of therenowned Greek logician Aristotle.

• Inductive reasoning involves generalizing after apattern has been recognized and established; thistype of logic is used in the solving of puzzles.

WE’LL ANALYZE AND EXPLORE VARIOUS TYPESOF STATEMENTS AND THE CONDITIONS UNDERWHICH THEY ARE TRUE:

• A statement is a simple sentence that is either trueor false. Simple statements can be connected toform compound, or more complicated, statements.

• Symbolic representations reduce a compoundstatement to its basic form; phrases that appear tobe different may actually have the same basicstructure and meaning.

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Logic

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WHAT WE WILL DO In This Chapter — cont inued

WE’LL ANALYZE AND EXPLORE CONDITIONAL, OR “IF . . . THEN . . .,”STATEMENTS:

• In everyday conversation, we often connect phrases by saying “if this, thenthat.” However, does “this” actually guarantee “that”? Is “this” in factnecessary for “that”?

• How does “if” compare with “only if”? What does “if and only if” reallymean?

WE’LL DETERMINE THE VALIDITY OF AN ARGUMENT:

• What constitutes a valid argument? Can a valid argument yield a falseconclusion?

• You may have used Venn diagrams to depict a solution set in an algebraclass. We will use Venn diagrams to visualize and analyze an argument.

• Some of Lewis Carroll’s whimsical arguments are valid, and some are not.How can you tell?

Webster’s New World College Dictionary defines logic as “thescience of correct reasoning; science which describes relationships amongpropositions in terms of implication, contradiction, contrariety, conversion,etc.” In addition to being flaunted in Mr. Spock’s claim that “your humanemotions have drawn you to an illogical conclusion” and in SherlockHolmes’s immortal phrase “elementary, my dear Watson,” logic isfundamental both to critical thinking and to problem solving. In today’sworld of misleading commercial claims, innuendo, and political rhetoric,the ability to distinguish between valid and invalid arguments isimportant.

In this chapter, we will study the basic components of logic and itsapplication. Mischievous, wild-eyed residents of Wonderland, eccentric,violin-playing detectives, and cold, emotionless Vulcans are not the onlyones who can benefit from logic. Armed with the fundamentals of logic, wecan surely join Spock and “live long and prosper!”

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1.1 Deductive versus Inductive Reasoning 3

Logic is the science of correct reasoning.Auguste Rodin captured this ideal in hisbronze sculpture The Thinker.

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1.1 Deductive versus Inductive Reasoning

Objectives

• Use Venn diagrams to determine the validity of deductive arguments

• Use inductive reasoning to predict patterns

Logic is the science of correct reasoning. Webster’s New World College Dictionarydefines reasoning as “the drawing of inferences or conclusions from known or assumed facts.” Reasoning is an integral part of our daily lives; we take appropriateactions based on our perceptions and experiences. For instance, if the sky is heav-ily overcast this morning, you might assume that it will rain today and take yourumbrella when you leave the house.

Problem Solving

Logic and reasoning are associated with the phrases problem solving and criticalthinking. If we are faced with a problem, puzzle, or dilemma, we attempt to reasonthrough it in hopes of arriving at a solution.

The first step in solving any problem is to define the problem in a thoroughand accurate manner. Although this might sound like an obvious step, it is oftenoverlooked. Always ask yourself, “What am I being asked to do?” Before you can

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solve a problem, you must understand the question. Once the problem has beendefined, all known information that is relevant to it must be gathered, organized,and analyzed. This analysis should include a comparison of the present problem toprevious ones. How is it similar? How is it different? Does a previous method ofsolution apply? If it seems appropriate, draw a picture of the problem; visualrepresentations often provide insight into the interpretation of clues.

Before using any specific formula or method of solution, determine whetherits use is valid for the situation at hand. A common error is to use a formula ormethod of solution when it does not apply. If a past formula or method of solutionis appropriate, use it; if not, explore standard options and develop creative alterna-tives. Do not be afraid to try something different or out of the ordinary. “What if Itry this . . . ?” may lead to a unique solution.

Deductive Reasoning

Once a problem has been defined and analyzed, it might fall into a known categoryof problems, so a common method of solution may be applied. For instance, whenone is asked to solve the equation x2 � 2x � 1, realizing that it is a second-degreeequation (that is, a quadratic equation) leads one to put it into the standard form (x2 � 2x � 1 � 0) and apply the Quadratic Formula.

EXAMPLE 1 USING DEDUCTIVE REASONING TO SOLVE AN EQUATION Solve theequation x2 � 2x � 1.

SOLUTION The given equation is a second-degree equation in one variable. We know that allsecond-degree equations in one variable (in the form ax2 � bx � c � 0) can besolved by applying the Quadratic Formula:

x ��b ; 2b2 � 4ac

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4 CHAPTER 1 Logic

Using his extraordinary powers of logical deduction, Sherlock Holmessolves another mystery. “Finding the villain was elementary, my dearWatson.”

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Therefore, x2 � 2x � 1 can be solved by applying the Quadratic Formula:

The solutions are

In Example 1, we applied a general rule to a specific case; we reasoned thatit was valid to apply the (general) Quadratic Formula to the (specific) equation x2 � 2x � 1. This type of logic is known as deductive reasoning—that is, theapplication of a general statement to a specific instance.

Deductive reasoning and the formal structure of logic have been studied forthousands of years. One of the earliest logicians, and one of the most renowned,was Aristotle (384–322 B.C.). He was the student of the great philosopher Plato andthe tutor of Alexander the Great, the conqueror of all the land from Greece toIndia. Aristotle’s philosophy is pervasive; it influenced Roman Catholic theologythrough St. Thomas Aquinas and continues to influence modern philosophy. Forcenturies, Aristotelian logic was part of the education of lawyers and politiciansand was used to distinguish valid arguments from invalid ones.

For Aristotle, logic was the necessary tool for any inquiry, and the syllogismwas the sequence followed by all logical thought. A syllogism is an argument com-posed of two statements, or premises (the major and minor premises), followed bya conclusion. For any given set of premises, if the conclusion of an argument isguaranteed (that is, if it is inescapable in all instances), the argument is valid. If theconclusion is not guaranteed (that is, if there is at least one instance in which itdoes not follow), the argument is invalid.

Perhaps the best known of Aristotle’s syllogisms is the following:

1. All men are mortal. major premise2. Socrates is a man. minor premiseTherefore, Socrates is mortal. conclusion

When the major premise is applied to the minor premise, the conclusion isinescapable; the argument is valid.

Notice that the deductive reasoning used in the analysis of Example 1 hasexactly the same structure as Aristotle’s syllogism concerning Socrates:

1. All second-degree equations in one variable can be major premisesolved by applying the Quadratic Formula.

2. x2 � 2x � 1 is a second-degree equation in one variable. minor premiseTherefore, x2 � 2x � 1 can be solved by applying the conclusionQuadratic Formula.

x � 1 � 22 and x � 1 � 22.

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Each of these syllogisms is of the following general form:1. If A, then B. All A are B. (major premise)2. x is A. We have A. (minor premise)

Therefore, x is B. Therefore, we have B. (conclusion)Historically, this valid pattern of deductive reasoning is known as modus ponens.

Deductive Reasoning and Venn Diagrams

The validity of a deductive argument can be shown by use of a Venn diagram. AVenn diagram is a diagram consisting of various overlapping figures containedwithin a rectangle (called the “universe”). To depict a statement of the form “AllA are B” (or, equivalently, “If A, then B”), we draw two circles, one inside theother; the inner circle represents A, and the outer circle represents B. This rela-tionship is shown in Figure 1.1.

Venn diagrams depicting “No A are B” and “Some A are B” are shown in Fig-ures 1.2 and 1.3, respectively.

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EXAMPLE 2 ANALYZING A DEDUCTIVE ARGUMENT Construct a Venn diagram toverify the validity of the following argument:

1. All men are mortal.2. Socrates is a man.

Therefore, Socrates is mortal.

SOLUTION Premise 1 is of the form “All A are B” and can be represented by a diagram like thatshown in Figure 1.4.

Premise 2 refers to a specific man, namely, Socrates. If we let x � Socrates,the statement “Socrates is a man” can then be represented by placing x within thecircle labeled “men,” as shown in Figure 1.5. Because we placed x within the“men” circle, and all of the “men” circle is inside the “mortal” circle, the conclu-sion “Socrates is mortal” is inescapable; the argument is valid.

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supervise the educa-tion of his sonAlexander, the futureAlexander the Great.Aristotle acceptedthe invitation andtaught Alexanderuntil he succeededhis father as ruler. Atthat time, Aristotlefounded a schoolknown as the

Lyceum, or Peripatetic School. The schoolhad a large library with many maps, aswell as botanical gardens containing anextensive collection of plants and animals.Aristotle and his students would walkabout the grounds of the Lyceum while dis-cussing various subjects (peripatetic isfrom the Greek word meaning “to walk”).

Many consider Aristotle to be afounding father of the study of biologyand of science in general; he observedand classified the behavior andanatomy of hundreds of living creatures.Alexander the Great, during his manymilitary campaigns, had his troopsgather specimens from distant places forAristotle to study.

Aristotle was a prolific writer; somehistorians credit him with the writing ofover 1,000 books. Most of his workshave been lost or destroyed, but schol-ars have recreated some of his more in-fluential works, including Organon.

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Aristotle’s collective works on syllogisms anddeductive logic are known as Organon,meaning “instrument,” for logic is the instrumentused in the acquisition of knowledge.

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Aristotle was one of Plato’s brighteststudents; he frequently questioned Plato’steachings and openly disagreed withhim. Whereas Plato emphasized thestudy of abstract ideas and mathematicaltruth, Aristotle was more interested inobserving the “real world” around him.Plato often referred to Aristotle as “thebrain” or “the mind of the school.” Platocommented, “Where others need thespur, Aristotle needs the rein.”

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EXAMPLE 3 ANALYZING A DEDUCTIVE ARGUMENT Construct a Venn diagram todetermine the validity of the following argument:

1. All doctors are men.2. My mother is a doctor.

Therefore, my mother is a man.

SOLUTION Premise 1 is of the form “All A are B”; the argument is depicted in Figure 1.6.No matter where x is placed within the “doctors” circle, the conclusion “My

mother is a man” is inescapable; the argument is valid.

Saying that an argument is valid does not mean that the conclusion is true.The argument given in Example 3 is valid, but the conclusion is false. One’smother cannot be a man! Validity and truth do not mean the same thing. An argu-ment is valid if the conclusion is inescapable, given the premises. Nothing is saidabout the truth of the premises. Thus, when examining the validity of an argument,we are not determining whether the conclusion is true or false. Saying that anargument is valid merely means that, given the premises, the reasoning used toobtain the conclusion is logical. However, if the premises of a valid argument aretrue, then the conclusion will also be true.

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If x is placed as in Figure 1.7, the argument would appear to be valid; the fig-ure supports the conclusion “The Rock is a professional wrestler.” However, theplacement of x in Figure 1.8 does not support the conclusion; given the premises,we cannot logically deduce that “The Rock is a professional wrestler.” Since theconclusion is not inescapable, the argument is invalid.

Saying that an argument is invalid does not mean that the conclusion is false.Example 4 demonstrates that an invalid argument can have a true conclusion; eventhough The Rock is a professional wrestler, the argument used to obtain the con-clusion is invalid. In logic, validity and truth do not have the same meaning.Validity refers to the process of reasoning used to obtain a conclusion; truth refersto conformity with fact or experience.

8 CHAPTER 1 Logic

actors

U

x = The Rock

professionalwrestlers x

actors

U

x

x = The Rock

professionalwrestlers

FIGURE 1.7 FIGURE 1.8


1. All professional wrestlers are actors.2. The Rock is an actor.

Therefore, The Rock is a professional wrestler.

SOLUTION Premise 1 is of the form “All A are B”; the “circle of professional wrestlers” is con-tained within the “circle of actors.” If we let x represent The Rock, premise 2 sim-ply requires that we place x somewhere within the actor circle; x could be placedin either of the two locations shown in Figures 1.7 and 1.8.

Even though The Rock is aprofessional wrestler, theargument used to obtain theconclusion is invalid.

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1. Some plants are poisonous.2. Broccoli is a plant.

Therefore, broccoli is poisonous.

VENN DIAGRAMS AND INVALID ARGUMENTSTo show that an argument is invalid, you must construct a Venn diagram inwhich the premises are met yet the conclusion does not necessarily follow.

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of in Figure 1.8 does not support the conclusion; given the premises,

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conclusion is of conclusion is notof not inescapable, the argument is invalid.of inescapable, the argument is invalid.Cen

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is placed as in Figure 1.7, the argument would appear to be valid; the fig-ure supports the conclusion “The Rock is a professional wrestler.” However, theCen

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ure supports the conclusion “The Rock is a professional wrestler.” However, thein Figure 1.8 does not support the conclusion; given the premises,Cen

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in Figure 1.8 does not support the conclusion; given the premises,deduce that “The Rock is a professional wrestler.” Since theCen

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wrestlers


SOLUTION Premise 1 is of the form “Some A are B”; it can be represented by two overlappingcircles (as in Figure 1.3). If we let x represent broccoli, premise 2 requires that weplace x somewhere within the plant circle. If x is placed as in Figure 1.9, the argu-ment would appear to be valid. However, if x is placed as in Figure 1.10, the con-clusion does not follow. Because we can construct a Venn diagram in which thepremises are met yet the conclusion does not follow (Figure 1.10), the argument isinvalid.

plants

poison

U

x = broccoli

xplants

poison

U

x = broccoli

x

FIGURE 1.9 FIGURE 1.10

When analyzing an argument via a Venn diagram, you might have to drawthree or more circles, as in the next example.


1. No snake is warm-blooded.2. All mammals are warm-blooded.

Therefore, snakes are not mammals.

SOLUTION Premise 1 is of the form “No A are B”; it is depicted in Figure 1.11. Premise 2 is ofthe form “All A are B”; the “mammal circle” must be drawn within the “warm-blooded circle.” Both premises are depicted in Figure 1.12.

warm-blooded

snakes

U

mammalssnakes

warm-blooded

U

x = snake

x

No snake is warm-blooded.FIGURE 1.11 All mammals are warm-blooded.FIGURE 1.12

Because we placed x (� snake) within the “snake” circle, and the “snake”circle is outside the “warm-blooded” circle, x cannot be within the “mammal” circle(which is inside the “warm-blooded” circle). Given the premises, the conclusion“Snakes are not mammals” is inescapable; the argument is valid.

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of of Therefore, snakes are not mammals.

of Therefore, snakes are not mammals.

Premise 1 is of the form “No of Premise 1 is of the form “No

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ge When analyzing an argument via a Venn diagram, you might have to draw

Cenga

ge When analyzing an argument via a Venn diagram, you might have to draw

three or more circles, as in the next example.

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ANALYZING A DEDUCTIVE ARGUMENT

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ANALYZING A DEDUCTIVE ARGUMENTdetermine the validity of the following ar

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determine the validity of the following ar

1. No snake is warm-blooded.

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Therefore, snakes are not mammals.

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FIGURE 1.10

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When analyzing an argument via a Venn diagram, you might have to drawthree or more circles, as in the next example.

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three or more circles, as in the next example.

You might have encountered Venn diagrams when you studied sets in youralgebra class. The academic fields of set theory and logic are historically inter-twined; set theory was developed in the late nineteenth century as an aid in thestudy of logical arguments. Today, set theory and Venn diagrams are applied toareas other than the study of logical arguments; we will utilize Venn diagrams inour general study of set theory in Chapter 2.

Inductive Reasoning

The conclusion of a valid deductive argument (one that goes from general to spe-cific) is guaranteed: Given true premises, a true conclusion must follow. However,there are arguments in which the conclusion is not guaranteed even though thepremises are true. Consider the following:

1. Joe sneezed after petting Frako’s cat.2. Joe sneezed after petting Paulette’s cat.

Therefore, Joe is allergic to cats.

Is the conclusion guaranteed? If the premises are true, they certainly support theconclusion, but we cannot say with 100% certainty that Joe is allergic to cats. Theconclusion is not guaranteed. Maybe Joe is allergic to the flea powder that the catowners used; maybe he is allergic to the dust that is trapped in the cats’ fur; ormaybe he has a cold!

Reasoning of this type is called inductive reasoning. Inductive reasoninginvolves going from a series of specific cases to a general statement (see Figure 1.13).Although it may seem to follow and may in fact be true, the conclusion in an in-ductive argument is never guaranteed.

10 CHAPTER 1 Logic

general

specific

specific

general

Deductive Reasoning(Conclusion is guaranteed.)

Inductive Reasoning(Conclusion may be probable but is

not guaranteed.)

FIGURE 1.13

EXAMPLE 7 INDUCTIVE REASONING AND PATTERN RECOGNITION What is thenext number in the sequence 1, 8, 15, 22, 29, . . . ?

SOLUTION Noticing that the difference between consecutive numbers in the sequence is 7, wemay be tempted to say that the next term is 29 � 7 � 36. Is this conclusion guar-anteed? No! Another sequence in which numbers differ by 7 are dates of a givenday of the week. For instance, the dates of the Saturdays in the year 2011 are(January) 1, 8, 15, 22, 29, (February) 5, 12, 19, 26, . . . . Therefore, the next num-ber in the sequence 1, 8, 15, 22, 29, . . . might be 5. Without further information,we cannot determine the next number in the given sequence. We can only useinductive reasoning and give one or more possible answers.

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owners used; maybe he is allergic to the dust that is trapped in the cats’ fur; or

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owners used; maybe he is allergic to the dust that is trapped in the cats’ fur; or

Reasoning of this type is called inductive reasoning.

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Reasoning of this type is called inductive reasoning. involves going from a series of specific cases to a general statement (see Figure 1.13).

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involves going from a series of specific cases to a general statement (see Figure 1.13).Although it may seem to follow and may in fact be true,

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Although it may seem to follow and may in fact be true, ductive argument is never guaranteed.

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guaranteed. Maybe Joe is allergic to the flea powder that the cat

EXAMPLE 8 SOLVING A SUDOKU PUZZLE Solve the sudoku puzzle given in Figure 1.15.


Throughout history, people have al-ways been attracted to puzzles,

mazes, and brainteasers. Who candeny the inherent satisfaction of solvinga seemingly unsolvable or perplexingriddle? A popular new addition to theworld of puzzle solving is sudoku, anumbers puzzle. Loosely translatedfrom Japanese, sudoku means “singlenumber”; a sudoku puzzle simply in-volves placing the digits 1 through 9 in

a grid containing 9 rows and9 columns. In addition, the 9 by 9 gridof squares is subdivided into nine 3 by3 grids, or “boxes,” as shown in Fig-ure 1.14.

The rules of sudoku are quite simple:Each row, each column, and each boxmust contain the digits 1 through 9; andno row, column, or box can contain 2squares with the same number. Conse-quently, sudoku does not require any

arithmetic or mathematical skill; sudokurequires logic only. In solving a puzzle, acommon thought is “What happens if Iput this number here?”

Like crossword puzzles, sudoku puz-zles are printed daily in many newspa-pers across the country and around theworld. Web sites containing sudoku puz-zles and strategies provide an endlesssource of new puzzles and help. SeeExercise 62 to find links to popular sites.

Topic x

A blank sudoku grid.FIGURE 1.14

SUDOKU: LOGIC IN THE REAL WORLD

2

8

3

3

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7

8

6

4

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8

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1

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2

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1

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A sudoku puzzle.FIGURE 1.15

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zles are printed

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pers across the country and around the

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zles and strategies provide an endlesssource of new puzzles and help. See

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source of new puzzles and help. SeeExercise 62 to find links to popular sites.

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Exercise 62 to find links to popular sites.

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12 CHAPTER 1 Logic

1 2 34 5 67 8 9

2

6

1

1 2 3 4 5 6 7 8 9

2

3

4

5

6

7

8

9

8

7

Box numbers and coordinate system in sudoku.FIGURE 1.16

2

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The 6 in box 1 must be placed in square (3, 3).FIGURE 1.17

Examining boxes 1, 2, and 3, we see that boxes 2 and 3 each contain the digit5, whereas box 1 does not. We deduce that 5 must be placed in square (2, 3)because rows 1 and 3 already have a 5. In a similar fashion, square (1, 4) mustcontain 3. See Figure 1.18.

SOLUTION Recall that each 3 by 3 grid is referred to as a box. For convenience, the boxes arenumbered 1 through 9, starting in the upper left-hand corner and moving from leftto right, and each square can be assigned coordinates (x, y) based on its row num-ber x and column number y as shown in Figure 1.16.

For example, the digit 2 in Figure 1.16 is in box 1 and has coordinates (1, 3), thedigit 8 is in box 3 and has coordinates (1, 7), the digit 6 is in box 4 and has coordi-nates (5, 1) and the digit 7 is in box 9 and has coordinates (9, 7).

When you are first solving a sudoku puzzle, concentrate on only a few boxesrather than the puzzle as a whole. For instance, looking at boxes 1, 4, and 7, we seethat boxes 4 and 7 each contain the digit 6, whereas box 1 does not. Consequently,the 6 in box 1 must be placed in column 3 because (shaded) columns 1 and 2 al-ready have a 6. However, (shaded) row 2 already has a 6, so we can deduce that 6must be placed in row 3, column 3, that is, in square (3, 3) as shown in Figure 1.17.

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ready have a of ready have a 6 of 6. However, (shaded) row 2 already has a of . However, (shaded) row 2 already has a must be placed in row 3, column 3, that is, in square (3, 3) as shown in Figure 1.17.of must be placed in row 3, column 3, that is, in square (3, 3) as shown in Figure 1.17.

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For example, the digit 2 in Figure 1.16 is in box 1 and has coordinates (1, 3), the

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For example, the digit 2 in Figure 1.16 is in box 1 and has coordinates (1, 3), thedigit 8 is in box 3 and has coordinates (1, 7), the digit 6 is in box 4 and has coordi-

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digit 8 is in box 3 and has coordinates (1, 7), the digit 6 is in box 4 and has coordi-nates (5, 1) and the digit 7 is in box 9 and has coordinates (9, 7).

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nates (5, 1) and the digit 7 is in box 9 and has coordinates (9, 7).When you are first solving a sudoku puzzle, concentrate on only a few boxes

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When you are first solving a sudoku puzzle, concentrate on only a few boxesrather than the puzzle as a whole. For instance, looking at boxes 1, 4, and 7, we see

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rather than the puzzle as a whole. For instance, looking at boxes 1, 4, and 7, we seethat boxes 4 and 7 each contain the digit Cen

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that boxes 4 and 7 each contain the digit in box 1 must be placed in column 3 because (shaded) columns 1 and 2 al-Cen

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. However, (shaded) row 2 already has a

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7 8 977 8 9Box numbers and coordinate system in sudoku.Le

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Box numbers and coordinate system in sudoku.


Because we have placed two new digits in box 1, we might wish to focus onthe remainder of (shaded) box 1. Notice that the digit 4 can be placed only insquare (3, 1), as row 1 and column 2 already have a 4 in each of them; likewise, thedigit 9 can be placed only in square (1, 1) because column 2 already has a 9.Finally, either of the digits 1 or 7 can be placed in square (1, 2) or (3, 2) as shownin Figure 1.19. At some point later in the solution, we will be able to determine theexact values of squares (1, 2) and (3, 2), that is, which square receives a 1 andwhich receives a 7.

2

5

6

8

3

3

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2

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1

9

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7

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3

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5

8

6

Analyzing boxes 1, 2, and 3; placing the digits 3 and 5.FIGURE 1.18

2

5

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3

9

3

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1,7

8

1,7

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Focusing on box 1.FIGURE 1.19

Using this strategy of analyzing the contents of three consecutive boxes, wededuce the following placement of digits: 1 must go in (2, 5), 6 must go in (6, 7),6 must go in (8, 6), 7 must go in (8, 3), 3 must go in (8, 5), 8 must go in (9, 6), and5 must go in (7, 4). At this point, box 8 is complete as shown in Figure 1.20.(Remember, each box must contain each of the digits 1 through 9.)

Once again, we use the three consecutive box strategy and deduce the followingplacement of digits: 5 must go in (8, 7), 5 must go in (9, 1), 8 must go in (7, 1), 2 mustgo in (8, 1), and 1 must go in (7, 3). At this point, box 7 is complete as shown in Figure 1.21.

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1

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exact values of squares (1, 2) and (3, 2), that is, which square receives a 1 and

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Analyzing boxes 1, 2, and 3; placing the digits 3 and 5.

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Analyzing boxes 1, 2, and 3; placing the digits 3 and 5.

We now focus on box 4 and deduce the following placement of digits: 1 mustgo in (4, 1), 9 must go in (5, 3), 4 must go in (4, 3), 5 must go in (6, 2), 3 must goin (4, 2), and 2 must go in (5, 2). At this point, box 4 is complete. In addition, wededuce that 1 must go in (9, 8), and 3 must go in (5, 6) as shown in Figure 1.22.

14 CHAPTER 1 Logic

2

5

6

8

7

3

9

3

4

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7

1,7

8

1,7

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4

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3

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Box 8 is complete.FIGURE 1.20

2

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Once again, we use the three consecutive box strategy and deduce the follow-ing placement of digits: 3 must go in (6, 9), 3 must go in (7, 7), 9 must go in (2, 4),and 9 must go in (6, 5). Now, to finish row 6, we place 4 in (6, 8) and 2 in (6, 4) asshown in Figure 1.23. (Remember, each row must contain each of the digits 1through 9.)


After we place 7 in (5, 4), column 4 is complete. (Remember, each columnmust contain each of the digits 1 through 9.) This leads to placing 4 in (5, 5) and 8in (4, 5), thus completing box 5; row 5 is finalized by placing 8 in (5, 8) as shownin Figure 1.24.

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Row 6 is complete.FIGURE 1.23

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Column 4, box 5, and row 5 are complete.FIGURE 1.24

Now column 7 is completed by placing 4 in (2, 7) and 2 in (3, 7); placing 2 in(2, 6) and 7 in (3, 6) completes column 6 as shown in Figure 1.25.

At this point, we deduce that the digit in (3, 2) must be 1 because row 3 can-not have two 7’s. This in turn reveals that 7 must go in (1, 2), and box 1 is nowcomplete. To complete row 7, we place 4 in (7, 9) and 2 in (7, 8); row 4 is finishedwith 2 in (4, 9) and 7 in (4, 8). See Figure 1.26.

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As a final check, we scrutinize each box, row, and column to verify that nobox, row, or column contains the same digit twice. Congratulations, the puzzle hasbeen solved!

To finish rows 1, 2, and 3, 1 must go in (1, 9), 7 must go in (2, 9), and 9 mustgo in (3, 9). The puzzle is now complete as shown in Figure 1.27.

16 CHAPTER 1 Logic

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Box 1, row 7, and row 4 are complete.FIGURE 1.26

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Columns 7 and 6 are complete.FIGURE 1.25

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A completed sudoku puzzle.FIGURE 1.27

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17

1.1 Exercises

In Exercises 1–20, construct a Venn diagram to determine thevalidity of the given argument.

1. a. 1. All master photographers are artists.

2. Ansel Adams is a master photographer.

Therefore, Ansel Adams is an artist.

b. 1. All master photographers are artists.

2. Ansel Adams is an artist.

Therefore, Ansel Adams is a master photographer.

2. a. 1. All Olympic gold medal winners are rolemodels.

2. Michael Phelps is an Olympic gold medalwinner.

Therefore, Michael Phelps is a role model.

b. 1. All Olympic gold medal winners are rolemodels.

2. Michael Phelps is a role model.

Therefore, Michael Phelps is an Olympic gold medalwinner.

3. a. 1. All homeless people are unemployed.

2. Bill Gates is not a homeless person.

Therefore, Bill Gates is not unemployed.

b. 1. All homeless people are unemployed.

2. Bill Gates is not unemployed.

Therefore, Bill Gates is not a homeless person.

4. a. 1. All professional wrestlers are actors.

2. Ralph Nader is not an actor.

Therefore, Ralph Nader is not a professionalwrestler.

b. 1. All professional wrestlers are actors.

2. Ralph Nader is not a professional wrestler.

Therefore, Ralph Nader is not an actor.

5. 1. All pesticides are harmful to the environment.

2. No fertilizer is a pesticide.

Therefore, no fertilizer is harmful to the environment.

6. 1. No one who can afford health insurance isunemployed.

2. All politicians can afford health insurance.

Therefore, no politician is unemployed.

7. 1. No vegetarian owns a gun.

2. All policemen own guns.

Therefore, no policeman is a vegetarian.

8. 1. No professor is a millionaire.

2. No millionaire is illiterate.

Therefore, no professor is illiterate.

9. 1. All poets are loners.

2. All loners are taxi drivers.

Therefore, all poets are taxi drivers.

10. 1. All forest rangers are environmentalists.

2. All forest rangers are storytellers.

Therefore, all environmentalists are storytellers.

11. 1. Real men don’t eat quiche.

2. Clint Eastwood is a real man.

Therefore, Clint Eastwood doesn’t eat quiche.

12. 1. Real men don’t eat quiche.

2. Oscar Meyer eats quiche.

Therefore, Oscar Meyer isn’t a real man.

13. 1. All roads lead to Rome.

2. Route 66 is a road.

Therefore, Route 66 leads to Rome.

14. 1. All smiling cats talk.

2. The Cheshire Cat smiles.

Therefore, the Cheshire Cat talks.

15. 1. Some animals are dangerous.

2. A tiger is an animal.

Therefore, a tiger is dangerous.

16. 1. Some professors wear glasses.

2. Mr. Einstein wears glasses.

Therefore, Mr. Einstein is a professor.

17. 1. Some women are police officers.

2. Some police officers ride motorcycles.

Therefore, some women ride motorcycles.

18. 1. All poets are eloquent.

2. Some poets are wine connoisseurs.

Therefore, some wine connoisseurs are eloquent.

19. 1. All squares are rectangles.

2. Some quadrilaterals are squares.

Therefore, some quadrilaterals are rectangles.

20. 1. All squares are rectangles.

2. Some quadrilaterals are rectangles.

Therefore, some quadrilaterals are squares.

21. Classify each argument as deductive or inductive.

a. 1. My television set did not work two nights ago.

2. My television set did not work last night.

Therefore, my television set is broken.

b. 1. All electronic devices give their owners grief.

2. My television set is an electronic device.

Therefore, my television set gives me grief.

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18 CHAPTER 1 Logic

22. Classify each argument as deductive or inductive.

a. 1. I ate a chili dog at Joe’s and got indigestion.

2. I ate a chili dog at Ruby’s and got indigestion.

Therefore, chili dogs give me indigestion.

b. 1. All spicy foods give me indigestion.

2. Chili dogs are spicy food.

Therefore, chili dogs give me indigestion.

In Exercises 23–32, fill in the blank with what is most likely to bethe next number. Explain (using complete sentences) the patterngenerated by your answer.

23. 3, 8, 13, 18, _____

24. 10, 11, 13, 16, _____

25. 0, 2, 6, 12, _____

26. 1, 2, 5, 10, _____

27. 1, 4, 9, 16, _____

28. 1, 8, 27, 64, _____

29. 2, 3, 5, 7, 11, _____

30. 1, 1, 2, 3, 5, _____

31. 5, 8, 11, 2, _____

32. 12, 5, 10, 3, _____

In Exercises 33–36, fill in the blanks with what are most likely tobe the next letters. Explain (using complete sentences) the patterngenerated by your answers.

33. O, T, T, F, _____, _____

34. T, F, S, E, _____, _____

35. F, S, S, M, _____, _____

36. J, F, M, A, _____, _____

In Exercises 37–42, explain the general rule or pattern used toassign the given letter to the given word. Fill in the blank with theletter that fits the pattern.

37.

38.

39.

40.

41.

42.

43. Find two different numbers that could be used to fill inthe blank.

1, 4, 7, 10, _____

Explain the pattern generated by each of your answers.

44. Find five different numbers that could be used to fill inthe blank.

7, 14, 21, 28, ______

Explain the pattern generated by each of youranswers.

45. Example 1 utilized the Quadratic Formula. Verify that

is a solution of the equation ax2 � bx � c � 0.

HINT: Substitute the fraction for x in ax2 � bx � c andsimplify.

46. Example 1 utilized the Quadratic Formula. Verify that

is a solution of the equation ax2 � bx � c � 0.

HINT: Substitute the fraction for x in ax2 � bx � c andsimplify.

47. As a review of algebra, use the Quadratic Formula tosolve

x2 � 6x � 7 � 0

48. As a review of algebra, use the Quadratic Formula tosolve

x2 � 2x � 4 � 0

Solve the sudoku puzzles in Exercises 49–54.

49.

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circle square trapezoid octagon rectangle

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banana strawberry asparagus eggplant orange

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banana strawberry asparagus eggplant orange

y r g p _____

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1.1 Exercises 19

54.

Answer the following questions using completesentences and your own words.

• Concept Questions

55. Explain the difference between deductive reasoningand inductive reasoning.

56. Explain the difference between truth and validity.

57. What is a syllogism? Give an example of a syllogismthat relates to your life.

• History Questions

58. From the days of the ancient Greeks, the study oflogic has been mandatory in what two professions?Why?

59. Who developed a formal system of deductive logicbased on arguments?

60. What was the name of the school Aristotle founded?What does it mean?

61. How did Aristotle’s school of thought differ fromPlato’s?

Web Project

62. Obtain a sudoku puzzle and its solution from a popularweb site. Some useful links for this web project arelisted on the text web site:

www.cengage.com/math/johnson

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1.2 Symbolic Logic

Objectives

• Identify simple statements

• Express a compound statement in symbolic form

• Create the negation of a statement

• Express a conditional statement in terms of necessary and sufficient conditions

The syllogism ruled the study of logic for nearly 2,000 years and was not sup-planted until the development of symbolic logic in the late seventeenth century. Asits name implies, symbolic logic involves the use of symbols and algebraic manip-ulations in logic.

Statements

All logical reasoning is based on statements. A statement is a sentence that is either true or false.

EXAMPLE 1 IDENTIFYING STATEMENTS Which of the following are statements? Whyor why not?

a. Apple manufactures computers.b. Apple manufactures the world’s best computers.c. Did you buy a Dell?d. A $2,000 computer that is discounted 25% will cost $1,000.e. I am telling a lie.

SOLUTION a. The sentence “Apple manufactures computers” is true; therefore, it is a statement.b. The sentence “Apple manufactures the world’s best computers” is an opinion, and as

such, it is neither true nor false. It is true for some people and false for others. Therefore,it is not a statement.

c. The sentence “Did you buy a Dell?” is a question. As such, it is neither true nor false; itis not a statement.

d. The sentence “A $2,000 computer that is discounted 25% will cost $1,000” is false; there-fore, it is a statement. (A $2,000 computer that is discounted 25% would cost $1,500.)

e. The sentence “I am telling a lie” is a self-contradiction, or paradox. If it were true, thespeaker would be telling a lie, but in telling the truth, the speaker would be contradictingthe statement that he or she was lying; if it were false, the speaker would not be telling alie, but in not telling a lie, the speaker would be contradicting the statement that he or shewas lying. The sentence is not a statement.

By tradition, symbolic logic uses lowercase letters as labels for statements.The most frequently used letters are p, q, r, s, and t. We can label the statement “Itis snowing” as statement p in the following manner:

p: It is snowing.

If it is snowing, p is labeled true, whereas if it is not snowing, p is labeled false.20

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y I am telling a lie.

a.

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y a. The sentence “Apple manufactures computers” is true; therefore, it is a statement.

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y The sentence “Apple manufactures computers” is true; therefore, it is a statement.

b.

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IDENTIFYING STATEMENTS

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its name implies, symbolic logic involves the use of symbols and algebraic manip-

Compound Statements and Logical Connectives

It is easy to determine whether a statement such as “Charles donated blood” is trueor false; either he did or he didn’t. However, not all statements are so simple; someare more involved. For example, the truth of “Charles donated blood and did notwash his car, or he went to the library,” depends on the truth of the individualpieces that make up the larger, compound statement. A compound statement is astatement that contains one or more simpler statements. A compound statement canbe formed by inserting the word not into a simpler statement or by joining two ormore statements with connective words such as and, or, if . . . then . . . , only if, andif and only if. The compound statement “Charles did not wash his car” is formedfrom the simpler statement “Charles did wash his car.” The compound statement“Charles donated blood and did not wash his car, or he went to the library” consistsof three statements, each of which may be true or false.

Figure 1.28 diagrams two equivalent compound statements.

1.2 Symbolic Logic 21

out

in A

in B

in A

in B

Or Not out

“neither A nor B” “not A and not B”means

Not

Not

And

Technicians and engineers use compound statements and logicalconnectives to study the flow of electricity through switching circuits.

FIGURE 1.28

When is a compound statement true? Before we can answer this question, wemust first examine the various ways in which statements can be connected. Depend-ing on how the statements are connected, the resulting compound statement can bea negation, a conjunction, a disjunction, a conditional, or any combination thereof.

The Negation �p

The negation of a statement is the denial of the statement and is represented by thesymbol �. The negation is frequently formed by inserting the word not. For exam-ple, given the statement “p: It is snowing,” the negation would be “�p: It is notsnowing.” If it is snowing, p is true and �p is false. Similarly, if it is not snowing,p is false and �p is true. A statement and its negation always have opposite truthvalues; when one is true, the other is false. Because the truth of the negation de-pends on the truth of the original statement, a negation is classified as a compoundstatement.

EXAMPLE 2 WRITING A NEGATION Write a sentence that represents the negation of eachstatement:

a. The senator is a Democrat. b. The senator is not a Democrat.c. Some senators are Republicans. d. All senators are Republicans.e. No senator is a Republican.

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Figure 1.28 diagrams two equivalent compound statements.

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SOLUTION a. The negation of “The senator is a Democrat” is “The senator is not a Democrat.”b. The negation of “The senator is not a Democrat” is “The senator is a Democrat.”c. A common error would be to say that the negation of “Some senators are Republicans”

is “Some senators are not Republicans.” However, “Some senators are Republicans” isnot denied by “Some senators are not Republicans.” The statement “Some senators areRepublicans” implies that at least one senator is a Republican. The negation of this state-ment is “It is not the case that at least one senator is a Republican,” or (more commonlyphrased) the negation is “No senator is a Republican.”

d. The negation of “All senators are Republicans” is “It is not the case that all senators areRepublicans,” or “There exists a senator who is not a Republican,” or (more commonlyphrased) “Some senators are not Republicans.”

e. The negation of “No senator is a Republican” is “It is not the case that no senator is aRepublican” or, in other words, “There exists at least one senator who is a Republican.”If “some” is interpreted as meaning “at least one,” the negation can be expressed as“Some senators are Republicans.”

The words some, all, and no (or none) are referred to as quantifiers. Parts (c)through (e) of Example 2 contain quantifiers. The linked pairs of quantified state-ments shown in Figure 1.29 are negations of each other.

22 CHAPTER 1 Logic

All p are q.

Some p are q.

No p are q.

Some p are not q.

Negations of statements containing quantifiers.FIGURE 1.29

The Conjunction p ∧ q

Consider the statement “Norma Rae is a union member and she is a Democrat.”This is a compound statement, because it consists of two statements—“Norma Raeis a union member” and “she (Norma Rae) is a Democrat”—and the connectiveword and. Such a compound statement is referred to as a conjunction. A conjunc-tion consists of two or more statements connected by the word and. We use thesymbol ∧ to represent the word and; thus, the conjunction “p ∧ q” represents thecompound statement “p and q.”

EXAMPLE 3 TRANSLATING WORDS INTO SYMBOLS Using the symbolic represen-tations

p: Norma Rae is a union member.q: Norma Rae is a Democrat.

express the following compound statements in symbolic form:

a. Norma Rae is a union member and she is a Democrat.b. Norma Rae is a union member and she is not a Democrat.

SOLUTION a. The compound statement “Norma Rae is a union member and she is a Democrat” can berepresented as p ∧ q.

b. The compound statement “Norma Rae is a union member and she is not a Democrat”can be represented as p ∧ �q.

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The Conjunction

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quantifiers.through (e) of Example 2 contain quantifiers. The linked pairs of quantified state-

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through (e) of Example 2 contain quantifiers. The linked pairs of quantified state-ments shown in Figure 1.29 are negations of each other.

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ments shown in Figure 1.29 are negations of each other.

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The Disjunction p ∨ q

When statements are connected by the word or, a disjunction is formed. We usethe symbol ∨ to represent the word or. Thus, the disjunction “p ∨ q” represents thecompound statement “p or q.” We can interpret the word or in two ways. Considerthe statements

p: Kaitlin is a registered Republican.q: Paki is a registered Republican.

The statement “Kaitlin is a registered Republican or Paki is a registered Republi-can” can be symbolized as p ∨ q. Notice that it is possible that both Kaitlin andPaki are registered Republicans. In this example, or includes the possibility thatboth things may happen. In this case, we are working with the inclusive or.

Now consider the statements

p: Kaitlin is a registered Republican.q: Kaitlin is a registered Democrat.

The statement “Kaitlin is a registered Republican or Kaitlin is a registered Demo-crat” does not include the possibility that both may happen; one statement excludesthe other. When this happens, we are working with the exclusive or. In our studyof symbolic logic (as in most mathematics), we will always use the inclusive or.Therefore, “p or q” means “p or q or both.”

EXAMPLE 4 TRANSLATING SYMBOLS INTO WORDS Using the symbolic represen-tations

p: Juanita is a college graduate.q: Juanita is employed.

express the following compound statements in words:

a. p ∨ q b. p ∧ qc. p ∨ �q d. �p ∧ q

SOLUTION a. p ∨ q represents the statement “Juanita is a college graduate or Juanita is employed (orboth).”

b. p ∧ q represents the statement “Juanita is a college graduate and Juanita is employed.”c. p ∨ �q represents the statement “Juanita is a college graduate or Juanita is not

employed.”d. �p ∧ q represents the statement “Juanita is not a college graduate and Juanita is

employed.”

The Conditional p → q

Consider the statement “If it is raining, then the streets are wet.” This is a com-pound statement because it connects two statements, namely, “it is raining” and“the streets are wet.” Notice that the statements are connected with “if . . . then . . .”phrasing. Any statement of the form “if p then q” is called a conditional (or animplication); p is called the hypothesis (or premise) of the conditional, and q iscalled the conclusion of the conditional. The conditional “if p then q” is repre-sented by the symbols “p → q” (p implies q). When people use conditionals in


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Kaitlin is a registered Democrat.

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24 CHAPTER 1 Logic

GOTTFRIED WILHELM LEIBNIZ 1646–1716

Leibniz’s affinityfor logic was char-acterized by hissearch for a charac-teristica universalis,or “universal charac-ter.” Leibniz be-lieved that by com-bining logic andmathematics, a gen-eral symbolic lan-

guage could be created in which allscientific problems could be solved witha minimum of effort. In this universallanguage, statements and the logicalrelationships between them would berepresented by letters and symbols. In

In the early 1670s, Leibniz invented one of the world’s first mechanicalcalculating machines. Leibniz’s machine could multiply and divide, whereasan earlier machine invented by Blaise Pascal (see Chapter 3) could only addand subtract.

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Leibniz’s words, “All truths of reasonwould be reduced to a kind of calculus,and the errors would only be errors ofcomputation.” In essence, Leibniz be-lieved that once a problem had beentranslated into this universal language ofsymbolic logic, it would be solved auto-matically by simply applying the mathe-matical rules that governed the manipu-lation of the symbols.

Leibniz’s work in the field of symboliclogic did not arouse much academic cu-riosity; many say that it was too farahead of its time. The study of symboliclogic was not systematically investigatedagain until the nineteenth century.

In addition to cofounding calculus (see Chapter 13),

the German-born GottfriedWilhelm Leibniz contributedmuch to the development ofsymbolic logic. A precociouschild, Leibniz was self-taught inmany areas. He taught himselfLatin at the age of eight and beganthe study of Greek when he was twelve.In the process, he was exposed to thewritings of Aristotle and became in-trigued by formalized logic.

At the age of fifteen, Leibniz enteredthe University of Leipzig to study law. Hereceived his bachelor’s degree twoyears later, earned his master’s degreethe following year, and then transferredto the University of Nuremberg.

Leibniz received his doctorate in lawwithin a year and was immediatelyoffered a professorship but refused it, saying that he had “other things inmind.” Besides law, these “other things”included politics, religion, history, litera-ture, metaphysics, philosophy, logic,and mathematics. Thereafter, Leibnizworked under the sponsorship of thecourts of various nobles, serving aslawyer, historian, and librarian to theelite. At one point, Leibniz was offeredthe position of librarian at the Vaticanbut declined the offer.

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everyday speech, they often omit the word then, as in “If it is raining, the streetsare wet.” Alternatively, the conditional “if p then q” may be phrased as “q if p”(“The streets are wet if it is raining”).


p: I am healthy.q: I eat junk food.r: I exercise regularly.


a. I am healthy if I exercise regularly.b. If I eat junk food and do not exercise, then I am not healthy.

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logic did not arouse much academic cu-riosity; many say that it was too far

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logic was not systematically investigatedagain until the nineteenth century.

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again until the nineteenth century.


SOLUTION a. “I am healthy if I exercise regularly” is a conditional (if . . . then . . . ) and can be re-phrased as follows:

“ I exercise regularly, I am healthy.”

Statement r Statement pis the premise. is the conclusion.

The given compound statement can be expressed as r → p.b. “If I eat junk food and do not exercise, then I am not healthy” is a conditional

(if . . . then . . . ) that contains a conjunction (and) and two negations (not):

“If I eat junk food do exercise, then I am healthy.”

The premise contains The conclusiona conjunction and a negation. contains a negation.

The premise of the conditional can be represented by q ∧ �r, while the conclusioncan be represented by �p. Thus, the given compound statement has the symbolic form(q ∧ �r) → �p.

EXAMPLE 6 TRANSLATING WORDS INTO SYMBOLS Express the following statementsin symbolic form:

a. All mammals are warm-blooded.b. No snake is warm-blooded.

SOLUTION a. The statement “All mammals are warm-blooded” can be rephrased as “If it is a mam-mal, then it is warm-blooded.” Therefore, we define two simple statements p and q as

p: It is a mammal.

q: It is warm-blooded.

The statement now has the form

“ it is a mammal, it is warm-blooded.”

Statement p Statement qis the premise. is the conclusion.

and can be expressed as p → q. In general, any statement of the form “All p are q” canbe symbolized as p → q.

b. The statement “No snake is warm-blooded” can be rephrased as “If it is a snake, then itis not warm-blooded.” Therefore, we define two simple statements p and q as

p: It is a snake.

q: It is warm-blooded.

The statement now has the form

“ it is a snake, it is warm-blooded.”

Statement p The negation of statement qis the premise. is the conclusion.

and can be expressed as p → �q. In general, any statement of the form “No p is q” canbe symbolized as p → �q.

↑ ↑↑ ↑notthenIf

↑ ↑↑ ↑thenIf

↑ ↑↑ ↑notnotand


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All mammals are warm-blooded.

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All mammals are warm-blooded.No snake is warm-blooded.

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No snake is warm-blooded.

The statement “All mammals are warm-blooded” can be rephrased as “If it is a mam-

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mal, then it is warm-blooded.” Therefore, we define two simple statements

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It is a mammal.

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It is warm-blooded.

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a conjunction and a negation. contains a negation.

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a conjunction and a negation. contains a negation.

The premise of the conditional can be represented by

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The premise of the conditional can be represented by q

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q ∧

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∧ �

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�Thus, the given compound statement has the symbolic form

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26 CHAPTER 1 Logic

Necessary and Sufficient Conditions

As Example 6 shows, conditionals are not always expressed in the form “if p thenq.” In addition to “all p are q,” other standard forms of a conditional include state-ments that contain the word sufficient or necessary.

Consider the statement “Being a mammal is sufficient for being warm-blooded.” One definition of the word sufficient is “adequate.” Therefore, “being amammal” is an adequate condition for “being warm-blooded”; hence, “being a mam-mal” implies “being warm-blooded.” Logically, the statement “Being a mammal issufficient for being warm-blooded” is equivalent to saying “If it is a mammal, then itis warm-blooded.” Consequently, the general statement “p is sufficient for q” is analternative form of the conditional “if p then q” and can be symbolized as p → q.

“Being a mammal” is a sufficient (adequate) condition for “being warm-blooded,” but is it a necessary condition? Of course not: some animals are warm-blooded but are not mammals (chickens, for example). One definition of the wordnecessary is “required.” Therefore, “being a mammal” is not required for “beingwarm-blooded.” However, is “being warm-blooded” a necessary (required) condi-tion for “being a mammal”? Of course it is: all mammals are warm-blooded (that is,there are no cold-blooded mammals). Logically, the statement “being warm-bloodedis necessary for being a mammal” is equivalent to saying “If it is a mammal, then itis warm-blooded.” Consequently, the general statement “q is necessary for p” is analternative form of the conditional “if p then q” and can be symbolized as p → q.

In summary, a sufficient condition is the hypothesis or premise of a condi-tional statement, whereas a necessary condition is the conclusion of a conditionalstatement.

p: hypothesis p implies q q: conclusion


p: A person obeys the law.q: A person is arrested.


a. Being arrested is necessary for not obeying the law.b. Obeying the law is sufficient for not being arrested.

SOLUTION a. “Being arrested” is a necessary condition; hence, “a person is arrested” is the conclusionof a conditional statement.

A necessary condition.Therefore, the statement “Being arrested is necessary for not obeying the law” can berephrased as follows:

“ a person does obey the law, the person is arrested.”

The negation of statement p Statement q is theis the premise. conclusion.

The given compound statement can be expressed as �p → q.

↑ ↑↑ ↑thennotIf

A person is arrested.

A person doesnot obey the law.

Necessary conditionSufficient condition

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is warm-blooded.” Consequently, the general statement “

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alternative form of the conditional “if

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alternative form of the conditional “if p

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p then

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then q

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q” and can be symbolized as

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In summary, a sufficient condition is the hypothesis or premise of a condi-

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In summary, a sufficient condition is the hypothesis or premise of a condi-tional statement, whereas a necessary condition is the conclusion of a conditional

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tional statement, whereas a necessary condition is the conclusion of a conditional

hypothesisCenga

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hypothesis

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” and can be symbolized as

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(adequate) condition for “being warm-

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condition? Of course not: some animals are warm-

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condition? Of course not: some animals are warm-blooded but are not mammals (chickens, for example). One definition of the word

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blooded but are not mammals (chickens, for example). One definition of the wordis “required.” Therefore, “being a mammal” is not required for “being

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is “required.” Therefore, “being a mammal” is not required for “beingwarm-blooded.” However, is “being warm-blooded” a necessary (required) condi-

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warm-blooded.” However, is “being warm-blooded” a necessary (required) condi-tion for “being a mammal”? Of course it is: all mammals

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b. “Obeying the law” is a sufficient condition; hence, “A person obeys the law” is thepremise of a conditional statement.

A sufficient condition.

Therefore, the statement “Obeying the law is sufficient for not being arrested” can berephrased as follows:

“ a person obeys the law, the person is arrested.”

Statement p The negation of statement qis the premise. is the conclusion.

The given compound statement can be expressed as p → �q.

↑ ↑↑ ↑notthenIf

A person is not arrested.A person obeys the law.

1.2 Exercises 27

Statement Symbol Read as . . .

negation �p not p

conjunction p ∧ q p and q

disjunction p ∨ q p or q

conditional p → q if p, then q(implication) p is sufficient for q

q is necessary for p

Logical connectives.FIGURE 1.30

We have seen that a statement is a sentence that is either true or false and thatconnecting two or more statements forms a compound statement. Figure 1.30 sum-marizes the logical connectives and symbols that were introduced in this section.The various connectives have been defined; we can now proceed in our analysis ofthe conditions under which a compound statement is true. This analysis is carriedout in the next section.

1.2 Exercises

1. Which of the following are statements? Why or whynot?

a. George Washington was the first president of theUnited States.

b. Abraham Lincoln was the second president of theUnited States.

c. Who was the first vice president of the UnitedStates?

d. Abraham Lincoln was the best president.

2. Which of the following are statements? Why or whynot?

a. 3 � 5 � 6

b. Solve the equation 2x � 5 � 3.

c. x2 � 1 � 0 has no solution.

d. x2 � 1 � (x � 1)(x � 1)

e. Is a rational number?12

3. Determine which pairs of statements are negations ofeach other.

a. All of the fruits are red.

b. None of the fruits is red.

c. Some of the fruits are red.

d. Some of the fruits are not red.


a. Some of the beverages contain caffeine.

b. Some of the beverages do not contain caffeine.

c. None of the beverages contain caffeine.

d. All of the beverages contain caffeine.

5. Write a sentence that represents the negation of eachstatement.

a. Her dress is not red.

b. Some computers are priced under $100.

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connecting two or more statements forms a compound statement. Figure 1.30 sum-marizes the logical connectives and symbols that were introduced in this section.

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the conditions under which a compound statement is true. This analysis is carried

f. Riding a motorcycle or playing the guitar is neces-sary for wearing a leather jacket.

10. Using the symbolic representations

p: The car costs $70,000.

q: The car goes 140 mph.

r: The car is red.

express the following compound statements in sym-bolic form.

a. All red cars go 140 mph.

b. The car is red, goes 140 mph, and does not cost$70,000.

c. If the car does not cost $70,000, it does not go140 mph.

d. The car is red and it does not go 140 mph or cost$70,000.

e. Being able to go 140 mph is sufficient for a car tocost $70,000 or be red.

f. Not being red is necessary for a car to cost $70,000and not go 140 mph.

In Exercises 11–34, translate the sentence into symbolic form. Besure to define each letter you use. (More than one answer ispossible.)

11. All squares are rectangles.

12. All people born in the United States are Americancitizens.

13. No square is a triangle.

14. No convicted felon is eligible to vote.

15. All whole numbers are even or odd.

16. All muscle cars from the Sixties are polluters.

17. No whole number is greater than 3 and less than 4.

18. No electric-powered car is a polluter.

19. Being an orthodontist is sufficient for being a dentist.

20. Being an author is sufficient for being literate.

21. Knowing Morse code is necessary for operating atelegraph.

22. Knowing CPR is necessary for being a paramedic.

23. Being a monkey is sufficient for not being an ape.

24. Being a chimpanzee is sufficient for not being amonkey.

25. Not being a monkey is necessary for being an ape.

26. Not being a chimpanzee is necessary for being amonkey.

27. I do not sleep soundly if I drink coffee or eatchocolate.

28. I sleep soundly if I do not drink coffee or eatchocolate.

29. Your check is not accepted if you do not have a driver’slicense or a credit card.

30. Your check is accepted if you have a driver’s license ora credit card.

31. If you drink and drive, you are fined or you go to jail.

c. All dogs are four-legged animals.

d. No sleeping bag is waterproof.


a. She is not a vegetarian.

b. Some elephants are pink.

c. All candy promotes tooth decay.

d. No lunch is free.


p: The lyrics are controversial.

q: The performance is banned.

express the following compound statements in sym-bolic form.a. The lyrics are controversial, and the performance is

banned.b. If the lyrics are not controversial, the performance

is not banned.c. It is not the case that the lyrics are controversial or

the performance is banned.d. The lyrics are controversial, and the performance is

not banned.e. Having controversial lyrics is sufficient for banning

a performance.f. Noncontroversial lyrics are necessary for not ban-

ning a performance.


p: The food is spicy.

q: The food is aromatic.

express the following compound statements in sym-bolic form.a. The food is aromatic and spicy.b. If the food isn’t spicy, it isn’t aromatic.c. The food is spicy, and it isn’t aromatic.d. The food isn’t spicy or aromatic.

e. Being nonaromatic is sufficient for food to benonspicy.

f. Being spicy is necessary for food to be aromatic.


p: A person plays the guitar.

q: A person rides a motorcycle.

r: A person wears a leather jacket.

express the following compound statements in sym-bolic form.

a. If a person plays the guitar or rides a motorcycle,then the person wears a leather jacket.

b. A person plays the guitar, rides a motorcycle, andwears a leather jacket.

c. A person wears a leather jacket and doesn’t play theguitar or ride a motorcycle.

d. All motorcycle riders wear leather jackets.e. Not wearing a leather jacket is sufficient for not

playing the guitar or riding a motorcycle.

28 CHAPTER 1 Logic

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43. What is a negation? 44. What is a conjunction?45. What is a disjunction? 46. What is a conditional?47. What is a sufficient condition?48. What is a necessary condition?49. What is the difference between the inclusive or and the

exclusive or?50. Create a sentence that is a self-contradiction, or para-

dox, as in part (e) of Example 1.


51. In what academic field did Gottfried Leibniz receive hisdegrees? Why is the study of logic important in this field?

52. Who developed a formal system of logic based on syl-logistic arguments?

53. What is meant by characteristica universalis? Whoproposed this theory?

Exercises 54–58 refer to the following: A culinary institute has asmall restaurant in which the students prepare various dishes.The menu changes daily, and during a specific week, the followingdishes are to be prepared: moussaka, pilaf, quiche, ratatouille,stroganoff, and teriyaki. During the week, the restaurant doesnot prepare any other kind of dish. The selection of dishes therestaurant offers is consistent with the following conditions:

• If the restaurant offers pilaf, then it does not offer ratatouille.

• If the restaurant does not offer stroganoff, then it offers pilaf.

• If the restaurant offers quiche, then it offers both ratatouille andteriyaki.

• If the restaurant offers teriyaki, then it offers moussaka orstroganoff or both.

54. Which one of the following could be a complete andaccurate list of the dishes the restaurant offers on aspecific day?

a. pilaf, quiche, ratatouille, teriyaki

b. quiche, stroganoff, teriyaki

THE NEXT LEVELIf a person wants to pursue an advanced degree(something beyond a bachelor’s or four-yeardegree), chances are the person must take a stan-dardized exam to gain admission to a graduateschool or to be admitted into a specific program.These exams are intended to measure verbal,quantitative, and analytical skills that have devel-oped throughout a person’s life. Many classes andstudy guides are available to help people preparefor the exams. The following questions are typicalof those found in the study guides.

32. If you are rich and famous, you have many friends andenemies.

33. You get a refund or a store credit if the product isdefective.

34. The streets are slippery if it is raining or snowing.

35. Using the symbolic representationsp: I am an environmentalist.q: I recycle my aluminum cans.express the following in words.

a. p ∧ q b. p → q

c. �q → �p d. q ∨ �p

36. Using the symbolic representationsp: I am innocent.q: I have an alibi.express the following in words.

a. p ∧ q b. p → q

c. �q → �p d. q ∨ �p

37. Using the symbolic representationsp: I am an environmentalist.q: I recycle my aluminum cans.r: I recycle my newspapers.express the following in words.

a. (q ∨ r) → p b. �p → �(q ∨ r)

c. (q ∧ r) ∨ �p d. (r ∧ �q) → �p

38. Using the symbolic representationsp: I am innocent.q: I have an alibi.r: I go to jail.express the following in words.

a. (p ∨ q) → �r b. (p ∧ �q) → r

c. (�p ∧ q) ∨ r d. (p ∧ r) → �q

39. Which statement, #1 or #2, is more appropriate?Explain why.Statement #1: “Cold weather is necessary for it tosnow.”Statement #2: “Cold weather is sufficient for it to snow.”

40. Which statement, #1 or #2, is more appropriate?Explain why.Statement #1: “Being cloudy is necessary for it to rain.”Statement #2: “Being cloudy is sufficient for it to rain.”

41. Which statement, #1 or #2, is more appropriate?Explain why.Statement #1: “Having 31 days in a month is necessaryfor it not to be February.”Statement #2: “Having 31 days in a month is sufficientfor it not to be February.”

42. Which statement, #1 or #2, is more appropriate?Explain why.Statement #1: “Being the Fourth of July is necessaryfor the U.S. Post Office to be closed.”Statement #2: “Being the Fourth of July is sufficientfor the U.S. Post Office to be closed.”

1.2 Exercises 29

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History Questions

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57. If the restaurant does not offer teriyaki, then which oneof the following must be true?

a. The restaurant offers pilaf.

b. The restaurant offers at most three different dishes.

c. The restaurant offers at least two different dishes.

d. The restaurant offers neither quiche nor ratatouille.

e. The restaurant offers neither quiche nor pilaf.

58. If the restaurant offers teriyaki, then which one of thefollowing must be false?

a. The restaurant does not offer moussaka.

b. The restaurant does not offer ratatouille.

c. The restaurant does not offer stroganoff.

d. The restaurant offers ratatouille but not quiche.

e. The restaurant offers ratatouille but not stroganoff.

c. quiche, ratatouille, teriyaki

d. ratatouille, stroganoff

e. quiche, ratatouille

55. Which one of the following cannot be a complete andaccurate list of the dishes the restaurant offers on a spe-cific day?

a. moussaka, pilaf, quiche, ratatouille, teriyaki

b. quiche, ratatouille, stroganoff, teriyaki

c. moussaka, pilaf, teriyaki

d. stroganoff, teriyaki

e. pilaf, stroganoff

56. Which one of the following could be the only kind ofdish the restaurant offers on a specific day?

a. teriyaki b. stroganoff

c. ratatouille d. quiche

e. moussaka

30 CHAPTER 1 Logic

1.3 Truth Tables

Objectives

• Construct a truth table for a compound statement

• Determine whether two statements are equivalent

• Apply De Morgan’s Laws

Suppose your friend Maria is a doctor, and you know that she is a Democrat. Ifsomeone told you, “Maria is a doctor and a Republican,” you would say that thestatement was false. On the other hand, if you were told, “Maria is a doctor or aRepublican,” you would say that the statement was true. Each of these state-ments is a compound statement—the result of joining individual statementswith connective words. When is a compound statement true, and when is itfalse? To answer these questions, we must examine whether the individualstatements are true or false and the manner in which the statements areconnected.

The truth value of a statement is the classification of the statement as trueor false and is denoted by T or F. For example, the truth value of the statement“Santa Fe is the capital of New Mexico” is T. (The statement is true.) In contrast,the truth value of “Memphis is the capital of Tennessee” is F. (The statement isfalse.)

A convenient way of determining whether a compound statement is true orfalse is to construct a truth table. A truth table is a listing of all possible combina-tions of the individual statements as true or false, along with the resulting truthvalue of the compound statement. As we will see, truth tables also allow us to dis-tinguish valid arguments from invalid arguments.

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Determine whether two statements are equivalent

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Suppose your friend Maria is a doctor, and you know that she is a Democrat. If

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The restaurant offers ratatouille but not stroganoff.

The Negation �p

The negation of a statement is the denial, or opposite, of the statement. (As wasstated in the previous section, because the truth value of the negation depends on thetruth value of the original statement, a negation can be classified as a compoundstatement.) To construct the truth table for the negation of a statement, we must firstexamine the original statement. A statement p may be true or false, as shown inFigure 1.31. If the statement p is true, the negation �p is false; if p is false, �p istrue. The truth table for the compound statement �p is given in Figure 1.32. Row 1of the table is read “�p is false when p is true.” Row 2 is read “�p is true whenp is false.”

The Conjunction p ∧ q

A conjunction is the joining of two statements with the word and. The compoundstatement “Maria is a doctor and a Republican” is a conjunction with the followingsymbolic representation:

p: Maria is a doctor.q: Maria is a Republican.p ∧ q: Maria is a doctor and a Republican.

The truth value of a compound statement depends on the truth values of theindividual statements that make it up. How many rows will the truth table for theconjunction p ∧ q contain? Because p has two possible truth values (T or F) andq has two possible truth values (T or F), we need four (2 · 2) rows in order to listall possible combinations of Ts and Fs, as shown in Figure 1.33.

For the conjunction p ∧ q to be true, the components p and q must both betrue; the conjunction is false otherwise. The completed truth table for the conjunc-tion p ∧ q is given in Figure 1.34. The symbols p and q can be replaced by any state-ments. The table gives the truth value of the statement “p and q,” dependent uponthe truth values of the individual statements “p” and “q.” For instance, row 3 is read“The conjunction p ∧ q is false when p is false and q is true.” The other rows areread in a similar manner.

The Disjunction p ∨ q

A disjunction is the joining of two statements with the word or. The compoundstatement “Maria is a doctor or a Republican” is a disjunction (the inclusive or)with the following symbolic representation:

p: Maria is a doctor.q: Maria is a Republican.p ∨ q: Maria is a doctor or a Republican.

Even though your friend Maria the doctor is not a Republican, the disjunction“Maria is a doctor or a Republican” is true. For a disjunction to be true, at least oneof the components must be true. A disjunction is false only when both componentsare false. The truth table for the disjunction p ∨ q is given in Figure 1.35.

1.3 Truth Tables 31

Truth values for a statement p.

FIGURE 1.31

p

T

F

1.

2.

Truth table for a negation �p.

FIGURE 1.32

p ~p

T F

F T

1.

2.

Truth values for two statements.

FIGURE 1.33

p q

T T

T F

F T

F F

1.

2.

3.

4.

Truth table for a conjunction p ∧ q.

FIGURE 1.34

p q p ∧ q

T T T

T F F

F T F

F F F

1.

2.

3.

4.

1.

2.

3.

4.

Truth table for a disjunction p ∨ q.

FIGURE 1.35

p q p ∨ q

T T T

T F T

F T T

F F F

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Maria is a Republican.

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Maria is a Republican.Maria is a doctor and a Republican.

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Maria is a doctor and a Republican.

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EXAMPLE 1 CONSTRUCTING A TRUTH TABLE Under what specific conditions is thefollowing compound statement true? “I have a high school diploma, or I have afull-time job and no high school diploma.”

SOLUTION First, we translate the statement into symbolic form, and then we construct thetruth table for the symbolic expression. Define p and q as

p: I have a high school diploma.q: I have a full-time job.

The given statement has the symbolic representation p ∨ (q ∧ �p).Because there are two letters, we need 2 � 2 � 4 rows. We need to insert a col-

umn for each connective in the symbolic expression p ∨ (q ∧ �p). As in algebra,we start inside any grouping symbols and work our way out. Therefore, we need acolumn for �p, a column for q ∧ �p, and a column for the entire expressionp ∨ (q ∧ �p), as shown in Figure 1.36.

32 CHAPTER 1 Logic

Required columns in the truth table.FIGURE 1.36

1.

2.

3.

4.

p q ~p q ∧ ~p p ∨ (q ∧ ~p)

T T

T F

F T

F F

In the �p column, fill in truth values that are opposite those for p. Next, theconjunction q ∧ �p is true only when both components are true; enter a T in row 3and Fs elsewhere. Finally, the disjunction p ∨ (q ∧ �p) is false only when bothcomponents p and (q ∧ �p) are false; enter an F in row 4 and Ts elsewhere. Thecompleted truth table is shown in Figure 1.37.

p q ~p q ∧ ~p p ∨ (q ∧ ~p)

T T F F T

T F F F T

F T T T T

F F T F F

1.

2.

3.

4.

Truth table for p ∨ (q ∧ �p).FIGURE 1.37

As is indicated in the truth table, the symbolic expression p ∨ (q ∧ �p) is trueunder all conditions except one: row 4; the expression is false when both p and qare false. Therefore, the statement “I have a high school diploma, or I have a full-time job and no high school diploma” is true in every case except when the speakerhas no high school diploma and no full-time job.

If the symbolic representation of a compound statement consists of twodifferent letters, its truth table will have 2 � 2 � 4 rows. How many rows are re-quired if a compound statement consists of three letters—say, p, q, and r? Becauseeach statement has two possible truth values (T and F), the truth table must contain2 � 2 � 2 � 8 rows. In general, each time a new statement is added, the number ofrows doubles.

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EXAMPLE 2 CONSTRUCTING A TRUTH TABLE Under what specific conditions is thefollowing compound statement true? “I own a handgun, and it is not the case thatI am a criminal or police officer.”

SOLUTION First, we translate the statement into symbolic form, and then we construct the truthtable for the symbolic expression. Define the three simple statements as follows:

p: I own a handgun.q: I am a criminal.r: I am a police officer.

The given statement has the symbolic representation p ∧ �(q ∨ r). Since there arethree letters, we need 23 � 8 rows. We start with three columns, one for each letter.To account for all possible combinations of p, q, and r as true or false, proceed asfollows:

1. Fill the first half (four rows) of column 1 with Ts and the rest with Fs, as shown in Figure 1.38(a).

2. In the next column, split each half into halves, the first half receiving Ts and the secondFs. In other words, alternate two Ts and two Fs in column 2, as shown in Fig-ure 1.38(b).

3. Again, split each half into halves; the first half receives Ts, and the second half receivesFs. Because we are dealing with the third (last) column, the Ts and Fs will alternate, asshown in Figure 1.38(c).

1.3 Truth Tables 33

p q r

T

T

T

T

F

F

F

F

p q r

T T

T T

T F

T F

F T

F T

F F

F F

p q r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

1.

2.

3.

4.

5.

6.

7.

8.

1.

2.

3.

4.

5.

6.

7.

8.

1.

2.

3.

4.

5.

6.

7.

8.

(a) (b) (c)

Truth values for three statements.FIGURE 1.38

(This process of filling the first half of the first column with Ts and the second halfwith Fs and then splitting each half into halves with blocks of Ts and Fs applies toall truth tables.)

We need to insert a column for each connective in the symbolic expressionp ∧ �(q ∨ r), as shown in Figure 1.39.

NUMBER OF ROWSIf a compound statement consists of n individual statements, each representedby a different letter, the number of rows required in its truth table is 2n.

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table for the symbolic expression. Define the three simple statements as follows:

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The given statement has the symbolic representation

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p, q,

34 CHAPTER 1 Logic


p q r q ∨ r ~(q ∨ r) p ∧ ~(q ∨ r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

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3.

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5.

6.

7.

8.

p q r q ∨ r ~(q ∨ r) p ∧ ~(q ∨ r)

T T T T F F

T T F T F F

T F T T F F

T F F F T T

F T T T F F

F T F T F F

F F T T F F

F F F F T F

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4.

5.

6.

7.

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p q r q ∨ r ~(q ∨ r) p ∧ ~(q ∨ r)

T T T T F

T T F T F

T F T T F

T F F F T

F T T T F

F T F T F

F F T T F

F F F F T

1.

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3.

4.

5.

6.

7.

8.

Truth values of the expressions q ∨ r and ~(q ∨ r).FIGURE 1.40

Truth table for p ∧ �(q ∨ r ).FIGURE 1.41

Now fill in the appropriate symbol in the column under q ∨ r. Enter F if bothq and r are false; enter T otherwise (that is, if at least one is true). In the �(q ∨ r)column, fill in truth values that are opposite those for q ∨ r, as in Figure 1.40.

The conjunction p ∧ �(q ∨ r) is true only when both p and �(q ∨ r) are true;enter a T in row 4 and Fs elsewhere. The truth table is shown in Figure 1.41.

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q ∨

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As indicated in the truth table, the expression p ∧ �(q ∨ r) is true only whenp is true and both q and r are false. Therefore, the statement “I own a handgun, andit is not the case that I am a criminal or police officer” is true only when the speakerowns a handgun, is not a criminal, and is not a police officer—in other words, thespeaker is a law-abiding citizen who owns a handgun.

The Conditional p → q

A conditional is a compound statement of the form “If p, then q” and is symbol-ized p → q. Under what circumstances is a conditional true, and when is it false?Consider the following (compound) statement: “If you give me $50, then I will giveyou a ticket to the ballet.” This statement is a conditional and has the followingrepresentation:

p: You give me $50.q: I give you a ticket to the ballet.p → q: If you give me $50, then I will give you a ticket to the ballet.

The conditional can be viewed as a promise: If you give me $50, then I willgive you a ticket to the ballet. Suppose you give me $50; that is, suppose p is true.I have two options: Either I give you a ticket to the ballet (q is true), or I do not (q isfalse). If I do give you the ticket, the conditional p → q is true (I have kept mypromise); if I do not give you the ticket, the conditional p → q is false (I have notkept my promise). These situations are shown in rows 1 and 2 of the truth table inFigure 1.42. Rows 3 and 4 require further analysis.

Suppose you do not give me $50; that is, suppose p is false. Whether or not Igive you a ticket, you cannot say that I broke my promise; that is, you cannot say thatthe conditional p → q is false. Consequently, since a statement is either true or false,the conditional is labeled true (by default). In other words, when the premise p of aconditional is false, it does not matter whether the conclusion q is true or false. Inboth cases, the conditional p → q is automatically labeled true, because it is not false.

The completed truth table for a conditional is given in Figure 1.43. Noticethat the only circumstance under which a conditional is false is when the premisep is true and the conclusion q is false, as shown in row 2.

EXAMPLE 3 CONSTRUCTING A TRUTH TABLE Under what conditions is the symbolicexpression q → �p true?

SOLUTION Our truth table has 22 � 4 rows and contains a column for p, q, �p, and q → �p,as shown in Figure 1.44.

1.3 Truth Tables 35

p q p → q

T T T

T F F

F T ?

F F ?

1.

2.

3.

4.

What if p is false?

FIGURE 1.42

p q p → q

T T T

T F F

F T T

F F T

1.

2.

3.

4.

Truth table for p → q.

FIGURE 1.43

p q ~p q → ~p

T T

T F

F T

F F

1.

2.

3.

4.

p q ~p q → ~p

T T F F

T F F T

F T T T

F F T T

1.

2.

3.

4.

Required columns in thetruth table.

FIGURE 1.44 Truth table for q → �p.FIGURE 1.45

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following (compound) statement: “If you give me $50, then I will give

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you a ticket to the ballet.” This statement is a conditional and has the following

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give you a ticket to the ballet. Suppose you give me $50; that is, suppose

In the �p column, fill in truth values that are opposite those for p. Now, aconditional is false only when its premise (in this case, q) is true and its conclusion(in this case, �p) is false. Therefore, q → �p is false only in row 1; the conditionalq → �p is true under all conditions except the condition that both p and q are true.The completed truth table is shown in Figure 1.45.

EXAMPLE 4 CONSTRUCTING A TRUTH TABLE Construct a truth table for the follow-ing compound statement: “I walk up the stairs if I want to exercise or if the elevatorisn’t working.”

SOLUTION Rewriting the statement so the word if is first, we have “If I want to exercise or (if)the elevator isn’t working, then I walk up the stairs.”

Now we must translate the statement into symbols and construct a truth table.Define the following:

p: I want to exercise.q: The elevator is working.r: I walk up the stairs.

The statement now has the symbolic representation (p ∨ �q) → r. Because wehave three letters, our table must have 23 � 8 rows. Inserting a column for eachletter and a column for each connective, we have the initial setup shown in Figure 1.46.

36 CHAPTER 1 Logic

p q r ~q p ∨ ~q ( p ∨ ~q) → r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

1.

2.

3.

4.

5.

6.

7.

8.


In the column labeled �q, enter truth values that are the opposite of thoseof q. Next, enter the truth values of the disjunction p ∨ �q in column 5. Recall thata disjunction is false only when both components are false and is true otherwise.Consequently, enter Fs in rows 5 and 6 (since both p and �q are false) and Ts inthe remaining rows, as shown in Figure 1.47.

The last column involves a conditional; it is false only when its premise is trueand its conclusion is false. Therefore, enter Fs in rows 2, 4, and 8 (since p ∨ �q istrue and r is false) and Ts in the remaining rows. The truth table is shown inFigure 1.48.

As Figure 1.48 shows, the statement “I walk up the stairs if I want to exerciseor if the elevator isn’t working” is true in all situations except those listed in rows2, 4, and 8. For instance, the statement is false (row 8) when the speaker does notwant to exercise, the elevator is not working, and the speaker does not walk up thestairs—in other words, the speaker stays on the ground floor of the building whenthe elevator is broken.

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1.3 Truth Tables 37

Equivalent Expressions

When you purchase a car, the car is either new or used. If a salesperson told you,“It is not the case that the car is not new,” what condition would the car be in? Thiscompound statement consists of one individual statement (“p: The car is new”) andtwo negations:

“It is not the case that the car is not new.”

� �p

Does this mean that the car is new? To answer this question, we will construct atruth table for the symbolic expression �(�p) and compare its truth values withthose of the original p. Because there is only one letter, we need 21 � 2 rows, asshown in Figure 1.49.

We must insert a column for �p and a column for �(�p). Now, �p has truthvalues that are opposite those of p, and �(�p) has truth values that are oppositethose of �p, as shown in Figure 1.50.

↑ ↑↑ ↑

p q r ~q p ∨ ~q ( p ∨ ~q) → r

T T T F T

T T F F T

T F T T T

T F F T T

F T T F F

F T F F F

F F T T T

F F F T T

1.

2.

3.

4.

5.

6.

7.

8.

Truth values of the expressions.FIGURE 1.47

Truth table for (p ∨ �q) → r.FIGURE 1.48

p q r ~q p ∨ ~q ( p ∨ ~q) → r

T T T F T T

T T F F T F

T F T T T T

T F F T T F

F T T F F T

F T F F F T

F F T T T T

F F F T T F

1.

2.

3.

4.

5.

6.

7.

8.

p

T

F

1.

2.

Truth values of p.

FIGURE 1.49

1.

2.

Truth table for �(�p).

FIGURE 1.50

p ~p ~(~p)

T F T

F T F

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Notice that the values in the column labeled �(�p) are identical to those inthe column labeled p. Whenever this happens, the expressions are said to beequivalent and may be used interchangeably. Therefore, the statement “It is notthe case that the car is not new” is equivalent in meaning to the statement “Thecar is new.”

Equivalent expressions are symbolic expressions that have identical truthvalues in each corresponding entry. The expression p � q is read “p is equiva-lent to q” or “p and q are equivalent.” As we can see in Figure 1.50, an expres-sion and its double negation are logically equivalent. This relationship can beexpressed as p � �(�p).

EXAMPLE 5 DETERMINING WHETHER STATEMENTS ARE EQUIVALENT Are thestatements “If I am a homeowner, then I pay property taxes” and “I am ahomeowner, and I do not pay property taxes” equivalent?

SOLUTION We begin by defining the statements:

p: I am a homeowner.q: I pay property taxes.p → q: If I am a homeowner, then I pay property taxes.p ∧ �q: I am a homeowner, and I do not pay property taxes.

The truth table contains 22 � 4 rows, and the initial setup is shown in Figure 1.51.Now enter the appropriate truth values under �q (the opposite of q). Because

the conjunction p ∧ �q is true only when both p and �q are true, enter a T in row 2and Fs elsewhere. The conditional p → q is false only when p is true and q is false;therefore, enter an F in row 2 and Ts elsewhere. The completed truth table is shownin Figure 1.52.

Because the entries in the columns labeled p ∧ �q and p → q are not thesame, the statements are not equivalent. “If I am a homeowner, then I payproperty taxes” is not equivalent to “I am a homeowner and I do not pay propertytaxes.”

38 CHAPTER 1 Logic

p q ~q p ∧ ~q p → q

T T

T F

F T

F F

1.

2.

3.

4.


FIGURE 1.51

p q ~q p ∧ ~q p → q

T T F F T

T F T T F

F T F F T

F F T F T

1.

2.

3.

4.

Truth table for p → q.FIGURE 1.52

Notice that the truth values in the columns under p ∧ �q and p → q in Fig-ure 1.52 are exact opposites; when one is T, the other is F. Whenever this happens,one statement is the negation of the other. Consequently, p ∧ �q is the negation ofp → q (and vice versa). This can be expressed as p ∧ �q � �(p → q). The nega-tion of a conditional is logically equivalent to the conjunction of the premise and thenegation of the conclusion.

Statements that look or sound different may in fact have the same meaning. Forexample, “It is not the case that the car is not new” really means the same as “The caris new,” and “It is not the case that if I am a homeowner, then I pay property taxes”

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1.3 Truth Tables 39

GEORGE BOOLE, 1815–1864

Boole’s most influen-tial work, An Investiga-tion of the Laws ofThought, on Which AreFounded the Mathe-matical Theories ofLogic and Probabilities,was published in1854. In it, he wrote,“There exist certain

general principles founded in the very na-ture of language and logic that exhibitlaws as identical in form as with the lawsof the general symbols of algebra.” Withthis insight, Boole had taken a big stepinto the world of logical reasoning andabstract mathematical analysis.

Perhaps because of his lack of formaltraining, Boole challenged the status quo,including the Aristotelian assumption thatall logical arguments could be reducedto syllogistic arguments. In doing so, heemployed symbols to represent concepts,as did Leibniz, but he also developedsystems of algebraic manipulation to ac-company these symbols. Thus, Boole’screation is a marriage of logic and math-ematics. However, as is the case with al-most all new theories, Boole’s symboliclogic was not met with total approbation.In particular, one staunch opponent of hiswork was Georg Cantor, whose work onthe origins of set theory and the magni-tude of infinity will be investigated inChapter 2.

In the many years since Boole’soriginal work was unveiled, variousscholars have modified, improved, gen-eralized, and extended its central con-cepts. Today, Boolean algebras are theessence of computer software and cir-cuit design. After all, a computer merelymanipulates predefined symbols andconforms to a set of preassigned alge-braic commands.

HistoricalNote

Through an algebraic manipulation of logicalsymbols, Boole revolutionized the age-old study oflogic. His essay The Mathematical Analysis ofLogic laid the foundation for his later bookAn Investigation of the Laws of Thought.

Brow

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sity

Lib

rary

George Boole is called “thefather of symbolic logic.”

Computer science owes muchto this self-educated mathemati-cian. Born the son of a poorshopkeeper in Lincoln, Eng-land, Boole had very little for-mal education, and his prospects for ris-ing above his family’s lower-class statuswere dim. Like Leibniz, he taught himselfLatin; at the age of twelve, he translatedan ode of Horace into English, winningthe attention of the local schoolmasters.(In his day, knowledge of Latin was aprerequisite to scholarly endeavors andto becoming a socially accepted gentle-man.) After that, his academic desireswere encouraged, and at the age of fif-teen, he began his long teaching ca-reer. While teaching arithmetic, he stud-ied advanced mathematics and physics.

In 1849, after nineteen years ofteaching at elementary schools, Boolereceived his big break: He was ap-pointed professor of mathematics atQueen’s College in the city of Cork,Ireland. At last, he was able to researchadvanced mathematics, and he be-came recognized as a first-class mathe-matician. This was a remarkable feat,considering Boole’s lack of formal train-ing and degrees.

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actually means the same as “I am a homeowner, and I do not pay property taxes.”When we are working with equivalent statements, we can substitute either state-ment for the other without changing the truth value.

De Morgan’s Laws

Earlier in this section, we saw that the negation of a negation is equivalent to theoriginal statement; that is, �(�p) � p. Another negation “formula” that we dis-covered was �(p → q) � p ∧ �q, that is, the negation of a conditional. Can wefind similar “formulas” for the negations of the other basic connectives, namely,the conjunction and the disjunction? The answer is yes, and the results are creditedto the English mathematician and logician Augustus De Morgan.

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40 CHAPTER 1 Logic

p ~p

T F

F T

p q p ∧ q

T T T

T F F

F T F

F F F

p q p ∨ q

T T T

T F T

F T T

F F F

p q p → q

T T T

T F F

F T T

F F T

Negation

Conjunction Disjunction Conditional

1.

2.

1.

2.

3.

4.

1.

2.

3.

4.

1.

2.

3.

4.

Truth tables for the basic connectives.FIGURE 1.53

De Morgan’s Laws are easily verified through the use of truth tables and willbe addressed in the exercises (see Exercises 55 and 56).

EXAMPLE 6 APPLYING DE MORGAN’S LAWS Using De Morgan’s Laws, find the negationof each of the following:

a. It is Friday and I receive a paycheck.b. You are correct or I am crazy.

SOLUTION a. The symbolic representation of “It is Friday and I receive a paycheck” is

p: It is Friday.

q: I receive a paycheck.

p ∧ q: It is Friday and I receive a paycheck.

Therefore, the negation is �(p ∧ q) � �p ∨ �q, that is, “It is not Friday or I do not re-ceive a paycheck.”

b. The symbolic representation of “You are correct or I am crazy” is

p: You are correct.

q: I am crazy.

p ∨ q: You are correct or I am crazy.

Therefore, the negation is �(p ∨ q) � �p ∧ �q, that is, “You are not correct and I amnot crazy.”

As we have seen, the truth value of a compound statement depends on thetruth values of the individual statements that make it up. The truth tables of thebasic connectives are summarized in Figure 1.53.

Equivalent statements are statements that have the same meaning. Equivalentstatements for the negations of the basic connectives are given in Figure 1.54.

DE MORGAN’S LAWSThe negation of the conjunction p ∧ q is given by �(p ∧ q) � �p ∨ �q.

“Not p and q” is equivalent to “not p or not q.”

The negation of the disjunction p ∨ q is given by �(p ∨ q) � �p ∧ �q.

“Not p or q” is equivalent to “not p and not q.”

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1.3 Exercises 41

1. ~(~p) ≡ p the negation of a negation2. ~( p ∧ q) ≡ ~p ∨ ~q the negation of a conjunction3. ~( p ∨ q) ≡ ~p ∧ ~q the negation of a disjunction4. ~( p → q) ≡ p ∧ ~q the negation of a conditional

Negations of the basic connectives.FIGURE 1.54

1.3 Exercises

In Exercises 1–20, construct a truth table for the symbolicexpressions.

1. p ∨ �q2. p ∧ �q

3. p ∨ �p

4. p ∧ �p

5. p → �q

6. �p → q

7. �q → �p

8. �p → �q

9. (p ∨ q) → �p

10. (p ∧ q) → �q

11. (p ∨ q) → (p ∧ q)

12. (p ∧ q) → (p ∨ q)

13. p ∧ �(q ∨ r)

14. p ∨ �(q ∨ r)

15. p ∨ (�q ∧ r)

16. �p ∨ �(q ∧ r)

17. (�r ∨ p) → (q ∧ p)

18. (q ∧ p) → (�r ∨ p)

19. (p ∨ r) → (q ∧ �r)

20. (p ∧ r) → (q ∨ �r)

In Exercises 21–40, translate the compound statement intosymbolic form and then construct the truth table for theexpression.

21. If it is raining, then the streets are wet.

22. If the lyrics are not controversial, the performance isnot banned.

23. The water supply is rationed if it does not rain.

24. The country is in trouble if he is elected.


26. All muscle cars from the Sixties are polluters.

27. No square is a triangle.

28. No electric-powered car is a polluter.


30. Being a chimpanzee is sufficient for not being a monkey.

31. Not being a monkey is necessary for being an ape.

32. Not being a chimpanzee is necessary for being a monkey.

33. Your check is accepted if you have a driver’s license ora credit card.

34. Yougeta refundorastorecredit if theproduct isdefective.

35. If leaded gasoline is used, the catalytic converter isdamaged and the air is polluted.

36. If he does not go to jail, he is innocent or has an alibi.

37. I have a college degree and I do not have a job or owna house.

38. I surf the Internet and I make purchases and do not paysales tax.

39. If Proposition A passes and Proposition B does not,jobs are lost or new taxes are imposed.

40. If Proposition A does not pass and the legislature raisestaxes, the quality of education is lowered and unem-ployment rises.

In Exercises 41–50, construct a truth table to determine whetherthe statements in each pair are equivalent.

41. The streets are wet or it is not raining.If it is raining, then the streets are wet.

42. The streets are wet or it is not raining.If the streets are not wet, then it is not raining.

43. He has a high school diploma or he is unemployed.If he does not have a high school diploma, then he isunemployed.

44. She is unemployed or she does not have a high schooldiploma.If she is employed, then she does not have a highschool diploma.

45. If handguns are outlawed, then outlaws have handguns.If outlaws have handguns, then handguns are outlawed.

46. If interest rates continue to fall, then I can afford to buya house.If interest rates do not continue to fall, then I cannotafford to buy a house.

47. If the spotted owl is on the endangered species list,then lumber jobs are lost.

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In Exercises 57–68, write the statement in symbolic form,construct the negation of the expression (in simplified symbolicform), and express the negation in words.

57. I have a college degree and I am not employed.

58. It is snowing and classes are canceled.

59. The television set is broken or there is a power outage.

60. The freeway is under construction or I do not ridethe bus.

61. If the building contains asbestos, the original contrac-tor is responsible.

62. If the legislation is approved, the public is uninformed.

63. The First Amendment has been violated if the lyricsare censored.

64. Your driver’s license is taken away if you do not obeythe laws.

65. Rainy weather is sufficient for not washing my car.

66. Drinking caffeinated coffee is sufficient for notsleeping.

67. Not talking is necessary for listening.

68. Not eating dessert is necessary for being on a diet.



69. a. Under what conditions is a disjunction true?b. Under what conditions is a disjunction false?

70. a. Under what conditions is a conjunction true?b. Under what conditions is a conjunction false?

71. a. Under what conditions is a conditional true?b. Under what conditions is a conditional false?

72. a. Under what conditions is a negation true?b. Under what conditions is a negation false?

73. What are equivalent expressions?

74. What is a truth table?

75. When constructing a truth table, how do you determinehow many rows to create?


76. Who is considered “the father of symbolic logic”?

77. Boolean algebra is a combination of logic and math-ematics. What is it used for?

If lumber jobs are not lost, then the spotted owl is noton the endangered species list.

48. If I drink decaffeinated coffee, then I do not stayawake.If I do stay awake, then I do not drink decaffeinatedcoffee.

49. The plaintiff is innocent or the insurance companydoes not settle out of court.The insurance company settles out of court and theplaintiff is not innocent.

50. The plaintiff is not innocent and the insurance com-pany settles out of court.It is not the case that the plaintiff is innocent or the in-surance company does not settle out of court.

In Exercises 51–54, construct truth tables to determine whichpairs of statements are equivalent.

51. i. Knowing Morse code is sufficient for operating atelegraph.

ii. Knowing Morse code is necessary for operating atelegraph.

iii. Not knowing Morse code is sufficient for notoperating a telegraph.

iv. Not knowing Morse code is necessary for notoperating a telegraph.

52. i. Knowing CPR is necessary for being a paramedic.

ii. Knowing CPR is sufficient for being a paramedic.

iii. Not knowing CPR is necessary for not being aparamedic.

iv. Not knowing CPR is sufficient for not being aparamedic.

53. i. The water being cold is necessary for not goingswimming.

ii. The water not being cold is necessary for goingswimming.

iii. The water being cold is sufficient for not goingswimming.

iv. The water not being cold is sufficient for goingswimming.

54. i. The sky not being clear is sufficient for it to beraining.

ii. The sky being clear is sufficient for it not to beraining.

iii. The sky not being clear is necessary for it to be raining.

iv. Theskybeingclear isnecessary for itnot toberaining.

55. Using truth tables, verify De Morgan’s Law

�(p ∧ q) � �p ∨ �q.

56. Using truth tables, verify De Morgan’s Law

�(p ∨ q) � �p ∧ �q.

42 CHAPTER 1 Logic

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1.4 More on Conditionals

Objectives

• Create the converse, inverse, and contrapositive of a conditional statement

• Determine equivalent variations of a conditional statement

• Interpret “only if” statements

• Interpret a biconditional statement

Conditionals differ from conjunctions and disjunctions with regard to the possibil-ity of changing the order of the statements. In algebra, the sum x � y is equal to thesum y � x; that is, addition is commutative. In everyday language, one realtor mightsay, “The house is perfect and the lot is priceless,” while another says, “The lot ispriceless and the house is perfect.” Logically, their meanings are the same, since(p ∧ q) � (q ∧ p). The order of the components in a conjunction or disjunction makesno difference in regard to the truth value of the statement. This is not so withconditionals.

Variations of a Conditional

Given two statements p and q, various “if . . . then . . .” statements can be formed.

EXAMPLE 1 TRANSLATING SYMBOLS INTO WORDS Using the statements

p: You are compassionate.q: You contribute to charities.

write an “if . . . then . . .” sentence represented by each of the following:

a. p → q b. q → p c. �p → �q d. �q → �p

SOLUTION a. p → q: If you are compassionate, then you contribute to charities.b. q → p: If you contribute to charities, then you are compassionate.c. �p → �q: If you are not compassionate, then you do not contribute to charities.d. �q → �p: If you do not contribute to charities, then you are not compassionate.

Each part of Example 1 contains an “if . . . then . . .” statement and is calleda conditional. Any given conditional has three variations: a converse, an inverse,and a contrapositive. The converse of the conditional “if p then q” is the com-pound statement “if q then p” That is, we form the converse of the conditional byinterchanging the premise and the conclusion; q → p is the converse of p → q. Thestatement in part (b) of Example 1 is the converse of the statement in part (a).

The inverse of the conditional “if p then q” is the compound statement “if notp then not q.” We form the inverse of the conditional by negating both the premiseand the conclusion; �p → �q is the inverse of p → q. The statement in part (c) ofExample 1 is the inverse of the statement in part (a).

The contrapositive of the conditional “if p then q” is the compoundstatement “if not q then not p.” We form the contrapositive of the conditional by

43

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negating and interchanging both the premise and the conclusion; �q → �p is thecontrapositive of p → q. The statement in part (d) of Example 1 is the contraposi-tive of the statement in part (a). The variations of a given conditional are summa-rized in Figure 1.55. As we will see, some of these variations are equivalent, andsome are not. Unfortunately, many people incorrectly treat them all as equivalent.

44 CHAPTER 1 Logic

Variations of a conditional.FIGURE 1.55

Name Symbolic Form Read As . . .

a (given) conditional p → q If p, then q.

the converse (of p → q) q → p If q, then p.

the inverse (of p → q) ~p → ~q If not p, then not q.

the contrapositive (of p → q) ~q → ~p If not q, then not p.

EXAMPLE 2 CREATING VARIATIONS OFA CONDITIONAL STATEMENT Given theconditional “You did not receive the proper refund if you prepared your ownincome tax form,” write the sentence that represents each of the following.

a. the converse of the conditionalb. the inverse of the conditionalc. the contrapositive of the conditional

SOLUTION a. Rewriting the statement in the standard “if . . . then . . .” form, we have the conditional“If you prepared your own income tax form, then you did not receive the proper re-fund.” The converse is formed by interchanging the premise and the conclusion. Thus,the converse is written as “If you did not receive the proper refund, then you preparedyour own income tax form.”

b. The inverse is formed by negating both the premise and the conclusion. Thus, the in-verse is written as “If you did not prepare your own income tax form, then you receivedthe proper refund.”

c. The contrapositive is formed by negating and interchanging the premise and the con-clusion. Thus, the contrapositive is written as “If you received the proper refund, thenyou did not prepare your own income tax form.”

Equivalent Conditionals

We have seen that the conditional p → q has three variations: the converse q → p, theinverse �p → �q, and the contrapositive �q → �p. Do any of these “if . . . then . . .”statements convey the same meaning? In other words, are any of these compoundstatements equivalent?

EXAMPLE 3 DETERMINING EQUIVALENT STATEMENTS Determine which (if any)of the following are equivalent: a conditional p → q, the converse q → p, theinverse �p → �q, and the contrapositive �q → �p.

SOLUTION To investigate the possible equivalencies, we must construct a truth table that con-tains all the statements. Because there are two letters, we need 22 � 4 rows. Thetable must have a column for �p, one for �q, one for the conditional p → q, andone for each variation of the conditional. The truth values of the negations �p and�q are readily entered, as shown in Figure 1.56.

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1.4 More on Conditionals 45

p q ~p ~q p → q q → p ~p → ~q ~q → ~p

T T F F

T F F T

F T T F

F F T T

1.

2.

3.

4.


An “if . . . then . . .” statement is false only when the premise is true and theconclusion is false. Consequently, p → q is false only when p is T and q is F; enteran F in row 2 and Ts elsewhere in the column under p → q.

Likewise, the converse q → p is false only when q is T and p is F; enter an Fin row 3 and Ts elsewhere.

In a similar manner, the inverse �p → �q is false only when �p is T and �qis F; enter an F in row 3 and Ts elsewhere.

Finally, the contrapositive �q → �p is false only when �q is T and �p is F;enter an F in row 2 and Ts elsewhere.

The completed truth table is shown in Figure 1.57. Examining the entries inFigure 1.57, we can see that the columns under p → q and �q → �p are identical;each has an F in row 2 and Ts elsewhere. Consequently, a conditional and its con-trapositive are equivalent: p → q � �q → �p.

Likewise, we notice that q → p and �p → �q have identical truth values;each has an F in row 3 and Ts elsewhere. Thus, the converse and the inverse of aconditional are equivalent: q → p � �p → �q.

p q ~p ~q p → q q → p ~p → ~q ~q → ~p

T T F F T T T T

T F F T F T T F

F T T F T F F T

F F T T T T T T

1.

2.

3.

4.

Truth table for a conditional and its variations.FIGURE 1.57

We have seen that different “if . . . then . . .” statements can convey the samemeaning—that is, that certain variations of a conditional are equivalent (see Fig-ure 1.58). For example, the compound statements “If you are compassionate, thenyou contribute to charities” and “If you do not contribute to charities, then you arenot compassionate” convey the same meaning. (The second conditional is the con-trapositive of the first.) Regardless of its specific contents (p, q, �p, or �q), every“if . . . then . . .” statement has an equivalent variation formed by negating and in-terchanging the premise and the conclusion of the given conditional statement.

Equivalent Statements Symbolic Representations

a conditional and its contrapositive ( p → q) ≡ (~q → ~p)

the converse and the inverse (of (q → p) ≡ (~p → ~q)the conditional p → q)

Equivalent “if . . . then . . .” statements.FIGURE 1.58

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46 CHAPTER 1 Logic

EXAMPLE 4 CREATING A CONTRAPOSITIVE Given the statement “Being a doctor isnecessary for being a surgeon,” express the contrapositive in terms of thefollowing:

a. a sufficient conditionb. a necessary condition

SOLUTION a. Recalling that a necessary condition is the conclusion of a conditional, we can rephrasethe statement “Being a doctor is necessary for being a surgeon” as follows:

“ a person is a surgeon, the person is a doctor.”

The premise. A necessary condition isthe conclusion.

Therefore, by negating and interchanging the premise and conclusion, the contraposi-tive is

“ a person is a doctor, the person is a surgeon.”

The negation of The negation ofthe conclusion. the premise.

Recalling that a sufficient condition is the premise of a conditional, we can phrase thecontrapositive of the original statement as “Not being a doctor is sufficient for not beinga surgeon.”

b. From part (a), the contrapositive of the original statement is the conditional statement


The premise of The conclusion of the contrapositive. the contrapositive.

Because a necessary condition is the conclusion of a conditional, the contrapositiveof the (original) statement “Being a doctor is necessary for being a surgeon” can beexpressed as “Not being a surgeon is necessary for not being a doctor.”

The “Only If” Connective

Consider the statement “A prisoner is paroled only if the prisoner obeys the rules.”What is the premise, and what is the conclusion? Rather than using p and q (whichmight bias our investigation), we define

r: A prisoner is paroled.s: A prisoner obeys the rules.

The given statement is represented by “r only if s.” Now, “r only if s” means thatr can happen only if s happens. In other words, if s does not happen, then r does nothappen, or �s → �r. We have seen that �s → �r is equivalent to r → s. Conse-quently, “r only if s” is equivalent to the conditional r → s. The premise of thestatement “A prisoner is paroled only if the prisoner obeys the rules” is “A prisoneris paroled,” and the conclusion is “The prisoner obeys the rules.”

The conditional p → q can be phrased “p only if q.” Even though the word ifprecedes q, q is not the premise. Whatever follows the connective “only if” is theconclusion of the conditional.

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EXAMPLE 5 ANALYZING AN “ONLY IF” STATEMENT For the compound statement“You receive a federal grant only if your artwork is not obscene,” do thefollowing:

a. Determine the premise and the conclusion.b. Rewrite the compound statement in the standard “if . . . then . . .” form.c. Interpret the conditions that make the statement false.

SOLUTION a. Because the compound statement contains an “only if ” connective, the statement thatfollows “only i f ” is the conclusion of the conditional. The premise is “You receive afederal grant.” The conclusion is “Your artwork is not obscene.”

b. The given compound statement can be rewritten as “If you receive a grant, then yourartwork is not obscene.”

c. First we define the symbols.

p: You receive a federal grant.

q: Your artwork is obscene.

Then the statement has the symbolic representation p → �q. The truth table for p → �q is given in Figure 1.59.

The expression p → q is false under the conditions listed in row 1 (when p andq are both true). Therefore, the statement “You receive a federal grant only if yourartwork is not obscene” is false when an artist does receive a federal grant and theartist’s artwork is obscene.

The Biconditional p ↔ q

What do the words bicycle, binomial, and bilingual have in common? Each wordbegins with the prefix bi, meaning “two.” Just as the word bilingual means “twolanguages,” the word biconditional means “two conditionals.”

In everyday speech, conditionals often get “hooked together” in a circularfashion. For instance, someone might say, “If I am rich, then I am happy, and if Iam happy, then I am rich.” Notice that this compound statement is actually the con-junction (and) of a conditional (if rich, then happy) and its converse (if happy, thenrich). Such a statement is referred to as a biconditional. A biconditional is a state-ment of the form (p → q) ∧ (q → p) and is symbolized as p ↔ q. The symbol p ↔ qis read “p if and only if q” and is frequently abbreviated “p iff q.” A biconditionalis equivalent to the conjunction of two conversely related conditionals: p ↔ q �[(p → q) ∧ (q → p)].

In addition to the phrase “if and only if,” a biconditional can also be ex-pressed by using “necessary” and “sufficient” terminology. The statement “p issufficient for q” can be rephrased as “if p then q” (and symbolized as p → q),whereas the statement “p is necessary for q” can be rephrased as “if q then p” (andsymbolized as q → p). Therefore, the biconditional “p if and only if q” can also bephrased as “p is necessary and sufficient for q.”

EXAMPLE 6 ANALYZING A BICONDITIONAL STATEMENT Express the biconditional“A citizen is eligible to vote if and only if the citizen is at least eighteen years old”as the conjunction of two conditionals.

SOLUTION The given biconditional is equivalent to “If a citizen is eligible to vote, then thecitizen is at least eighteen years old, and if a citizen is at least eighteen years old,then the citizen is eligible to vote.”

1.4 More on Conditionals 47

p q ~q p → ~q

T T F F

T F T T

F T F T

F F T T

1.

2.

3.

4.

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FIGURE 1.59

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3. p: I watch television.

q: I do my homework.

4. p: He is an artist.

q: He is a conformist.

In Exercises 5–10, form (a) the inverse, (b) the converse, and (c)the contrapositive of the given conditional.

5. If you pass this mathematics course, then you fulfill agraduation requirement.

6. If you have the necessary tools, assembly time is lessthan thirty minutes.

7. The television set does not work if the electricity isturned off.

8. You do not win if you do not buy a lottery ticket.

In Exercises 1–2, using the given statements, write the sentencerepresented by each of the following.

a. p → q b. q → p

c. �p → �q d. �q → �p

e. Which of parts (a)–(d) are equivalent? Why?

1. p: She is a police officer.

q: She carries a gun.

2. p: I am a multimillion-dollar lottery winner.

q: I am a world traveler.

In Exercises 3–4, using the given statements, write the sentencerepresented by each of the following.

a. p → �q b. �q → p

c. �p → q d. q → �p

e. Which of parts (a)–(d) are equivalent? Why?

Under what circumstances is the biconditional p ↔ q true, and when is itfalse? To find the answer, we must construct a truth table. Utilizing the equivalencep ↔ q � [(p → q) ∧ (q → p)], we get the completed table shown in Figure 1.60.(Recall that a conditional is false only when its premise is true and its conclusionis false and that a conjunction is true only when both components are true.) We cansee that a biconditional is true only when the two components p and q have thesame truth value—that is, when p and q are both true or when p and q are bothfalse. On the other hand, a biconditional is false when the two components p and qhave opposite truth value—that is, when p is true and q is false or vice versa.

48 CHAPTER 1 Logic

p q p → q q → p ( p → q) ∧ (q → p)

T T T T T

T F F T F

F T T F F

F F T T T

1.

2.

3.

4.

Truth table for a biconditional p ↔ q.FIGURE 1.60

Many theorems in mathematics can be expressed as biconditionals. For example, when solving a quadratic equation, we have the following: “The equationax 2 � bx � c � 0 has exactly one solution if and only if the discriminant b 2 � 4ac � 0.” Recall that the solutions of a quadratic equation are

This biconditional is equivalent to “If the equation ax 2 � bx � c � 0 has exactly onesolution, then the discriminant b2 � 4ac � 0, and if the discriminant b2 � 4ac � 0,then the equation ax2 � bx � c � 0 has exactly one solution”—that is, one condi-tion implies the other.

1.4 Exercises

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1.4 Exercises 49

9. You are a vegetarian if you do not eat meat.

10. If chemicals are properly disposed of, the environmentis not damaged.

In Exercises 11–14, express the contrapositive of the givenconditional in terms of (a) a sufficient condition and (b) anecessary condition.

11. Being an orthodontist is sufficient for being a dentist.

12. Being an author is sufficient for being literate.



In Exercises 15–20, (a) determine the premise and conclusion,(b) rewrite the compound statement in the standard “if . . .then . . .” form, and (c) interpret the conditions that make thestatement false.

15. I take public transportation only if it is convenient.

16. I eat raw fish only if I am in a Japanese restaurant.

17. I buy foreign products only if domestic products arenot available.

18. I ride my bicycle only if it is not raining.

19. You may become a U.S. senator only if you are at leastthirty years old and have been a citizen for nine years.

20. You may become the president of the United Statesonly if you are at least thirty-five years old and wereborn a citizen of the United States.

In Exercises 21–28, express the given biconditional as theconjunction of two conditionals.

21. You obtain a refund if and only if you have a receipt.

22. We eat at Burger World if and only if Ju Ju’s Kitsch-Inn is closed.

23. The quadratic equation ax2 � bx � c � 0 has two dis-tinct real solutions if and only if b2 � 4ac � 0.

24. The quadratic equation ax2 � bx � c � 0 has complexsolutions iff b2 � 4ac � 0.

25. A polygon is a triangle iff the polygon has threesides.

26. A triangle is isosceles iff the triangle has two equalsides.

27. A triangle having a 90° angle is necessary and suffi-cient for a2 � b2 � c2.

28. A triangle having three equal sides is necessary andsufficient for a triangle having three equal angles.

In Exercises 29–36, translate the two statements into symbolicform and use truth tables to determine whether the statements areequivalent.

29. I cannot have surgery if I do not have health insurance.

If I can have surgery, then I do have health insurance.

30. If I am illiterate, I cannot fill out an application form.

I can fill out an application form if I am not illiterate.

31. If you earn less than $12,000 per year, you are eligiblefor assistance.

If you are not eligible for assistance, then you earn atleast $12,000 per year.

32. If you earn less than $12,000 per year, you are eligiblefor assistance.

If you earn at least $12,000 per year, you are noteligible for assistance.

33. I watch television only if the program is educational.I do not watch television if the program is noteducational.

34. I buy seafood only if the seafood is fresh.

If I do not buy seafood, the seafood is not fresh.

35. Being an automobile that is American-made is suffi-cient for an automobile having hardware that is notmetric.

Being an automobile that is not American-made isnecessary for an automobile having hardware that ismetric.

36. Being an automobile having metric hardware is suffi-cient for being an automobile that is not American-made.

Being an automobile not having metric hardware isnecessary for being an automobile that is American-made.

In Exercises 37–46, write an equivalent variation of the givenconditional.

37. If it is not raining, I walk to work.

38. If it makes a buzzing noise, it is not working properly.

39. It is snowing only if it is cold.

40. You are a criminal only if you do not obey the law.

41. You are not a vegetarian if you eat meat.

42. You are not an artist if you are not creative.

43. All policemen own guns.

44. All college students are sleep deprived.

45. No convicted felon is eligible to vote.

46. No man asks for directions.

In Exercises 47–52, determine which pairs of statements areequivalent.

47. i. If Proposition 111 passes, freeways are improved.

ii. If Proposition 111 is defeated, freeways are notimproved.

iii. If the freeways are improved, Proposition 111passes.

iv. If the freeways are not improved, Proposition 111does not pass.

48. i. If the Giants win, then I am happy.

ii. If I am happy, then the Giants win.

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Exercises 58–62 refer to the following: Assuming that a movie’spopularity is measured by its gross box office receipts, six recentlyreleased movies—M, N, O, P, Q, and R—are ranked from mostpopular (first) to least popular (sixth). There are no ties. Theranking is consistent with the following conditions:

• O is more popular than R.

• If N is more popular than O, then neither Q nor R is morepopular than P.

• If O is more popular than N, then neither P nor R is morepopular than Q.

• M is more popular than N, or else M is more popular than O,but not both.

58. Which one of the following could be the ranking of themovies, from most popular to least popular?

a. N, M, O, R, P, Q

b. P, O, M, Q, N, R

c. Q, P, R, O, M, N

d. O, Q, M, P, N, R

e. P, Q, N, O, R, M

59. If N is the second most popular movie, then which oneof the following could be true?

a. O is more popular than M.

b. Q is more popular than M.

c. R is more popular than M.

d. Q is more popular than P.

e. O is more popular than N.

60. Which one of the following cannot be the most popu-lar movie?

a. M

b. N

c. O

d. P

e. Q

61. If R is more popular than M, then which one of the fol-lowing could be true?

a. M is more popular than O.

b. M is more popular than Q.

c. N is more popular than P.

d. N is more popular than O.

e. N is more popular than R.

62. If O is more popular than P and less popular than Q,then which one of the following could be true?

a. M is more popular than O.

b. N is more popular than M.

c. N is more popular than O.

d. R is more popular than Q.

e. P is more popular than R.

iii. If the Giants lose, then I am unhappy.

iv. If I am unhappy, then the Giants lose.

49. i. I go to church if it is Sunday.

ii. I go to church only if it is Sunday.

iii. If I do not go to church, it is not Sunday.

iv. If it is not Sunday, I do not go to church.

50. i. I am a rebel if I do not have a cause.

ii. I am a rebel only if I do not have a cause.

iii. I am not a rebel if I have a cause.

iv. If I am not a rebel, I have a cause.

51. i. If line 34 is greater than line 29, I use Schedule X.

ii. If I use Schedule X, then line 34 is greater thanline 29.

iii. If I do not use Schedule X, then line 34 is notgreater than line 29.

iv. If line 34 is not greater than line 29, then I do notuse Schedule X.

52. i. If you answer yes to all of the above, then youcomplete Part II.

ii. If you answer no to any of the above, then you donot complete Part II.

iii. If you completed Part II, then you answered yes toall of the above.

iv. If you did not complete Part II, then you answeredno to at least one of the above.



53. What is a contrapositive?

54. What is a converse?

55. What is an inverse?

56. What is a biconditional?

57. How is an “if . . . then . . .” statement related to an“only if” statement?

THE NEXT LEVELIf a person wants to pursue an advanced degree(something beyond a bachelor’s or four-yeardegree), chances are the person must take a stan-dardized exam to gain admission to a graduateschool or to be admitted into a specific program.These exams are intended to measure verbal,quantitative, and analytical skills that have devel-oped throughout a person’s life. Many classes andstudy guides are available to help people preparefor the exams. The following questions are typicalof those found in the study guides.

50 CHAPTER 1 Logic

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of the following could be true?

51

1.5 Analyzing Arguments

Objectives

• Identify a tautology

• Use a truth table to analyze an argument

Lewis Carroll’s Cheshire Cat told Alice that he was mad (crazy). Alice then asked,“‘And how do you know that you’re mad?’ ‘To begin with,’ said the cat, ‘a dog’snot mad. You grant that?’ ‘I suppose so,’ said Alice. ‘Well, then,’ the cat went on,‘you see a dog growls when it’s angry, and wags its tail when it’s pleased. Now Igrowl when I’m pleased, and wag my tail when I’m angry. Therefore I’m mad!’”

Does the Cheshire Cat have a valid deductive argument? Does the conclusionfollow logically from the hypotheses? To answer this question, and others like it,we will utilize symbolic logic and truth tables to account for all possible combina-tions of the individual statements as true or false.

Valid Arguments

When someone makes a sequence of statements and draws some conclusion fromthem, he or she is presenting an argument. An argument consists of two compo-nents: the initial statements, or hypotheses, and the final statement, or conclusion.When presented with an argument, a listener or reader may ask, “Does this personhave a logical argument? Does his or her conclusion necessarily follow from thegiven statements?”

An argument is valid if the conclusion of the argument is guaranteed underits given set of hypotheses. (That is, the conclusion is inescapable in all instances.)For example, we used Venn diagrams in Section 1.1 to show the argument

“All men are mortal.Socrates is a man.

the hypotheses

Therefore, Socrates is mortal.” the conclusion

is a valid argument. Given the hypotheses, the conclusion is guaranteed. The termvalid does not mean that all the statements are true but merely that the conclusionwas reached via a proper deductive process. As shown in Example 3 of Section 1.1,the argument

“All doctors are men.My mother is a doctor.

the hypotheses

Therefore, my mother is a man.” the conclusion

is also a valid argument. Even though the conclusion is obviously false, the con-clusion is guaranteed, given the hypotheses.

The hypotheses in a given logical argument may consist of several interrelatedstatements, each containing negations, conjunctions, disjunctions, and conditionals.By joining all the hypotheses in the form of a conjunction, we can form a single con-ditional that represents the entire argument. That is, if an argument has n hypotheses(h1, h2, . . . , hn) and conclusion c, the argument will have the form “if (h1 and h2 . . . andhn), then c.”

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given statements?”

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not mad. You grant that?’ ‘I suppose so,’ said Alice. ‘Well, then,’ the cat went on,

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‘you see a dog growls when it’s angry, and wags its tail when it’s pleased. Now

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‘you see a dog growls when it’s angry, and wags its tail when it’s pleased. Now growl when I’m pleased, and wag my tail when I’m angry. Therefore I’m mad!’”

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growl when I’m pleased, and wag my tail when I’m angry. Therefore I’m mad!’”ire Cat have a valid deductive argument? Does the conclusion

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we will utilize symbolic logic and truth tables to account for all possible combina-tions of the individual statements as true or false.

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tions of the individual statements as true or false.

52 CHAPTER 1 Logic

Using a logical argument, Lewis Carroll’s Cheshire Cat tried toconvince Alice that he was crazy. Was his argument valid?

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If the conditional representation of an argument is always true (regardlessof the actual truthfulness of the individual statements), the argument is valid. Ifthere is at least one instance in which the conditional is false, the argument isinvalid.

EXAMPLE 1 USING A TRUTH TABLE TO ANALYZE AN ARGUMENT Determinewhether the following argument is valid:

“If he is illiterate, he cannot fill out the application.He can fill out the application.Therefore, he is not illiterate.”

SOLUTION First, number the hypotheses and separate them from the conclusion with a line:

1. If he is illiterate, he cannot fill out the application.2. He can fill out the application.

Therefore, he is not illiterate.

Now use symbols to represent each different component in the statements:

p: He is illiterate.q: He can fill out the application.

CONDITIONAL REPRESENTATION OF AN ARGUMENTAn argument having n hypotheses h1, h2, . . . , hn and conclusion c can berepresented by the conditional [h1 ∧ h2 ∧ . . . ∧ hn] → c.

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We could have defined q as “He cannot fill out the application” (as stated in prem-ise 1), but it is customary to define the symbols with a positive sense. Symboli-cally, the argument has the form

1. p → �q

2. qthe hypotheses

∴ �p conclusion

and is represented by the conditional [(p → �q) ∧ q] → �p. The symbol ∴ is read“therefore.”

To construct a truth table for this conditional, we need 22 � 4 rows. A columnis required for the following: each negation, each hypothesis, the conjunction ofthe hypotheses, the conclusion, and the conditional representation of the argument.The initial setup is shown in Figure 1.61.

Fill in the truth table as follows:

�q: A negation has the opposite truth values; enter a T in rows 2 and 4 and an F inrows 1 and 3.

Hypothesis 1: A conditional is false only when its premise is true and its conclusionis false; enter an F in row 1 and Ts elsewhere.

⎫⎬⎭

⎫⎬⎭

1.5 Analyzing Arguments 53

Column ConditionalRepresenting Representation

Hypothesis Hypothesis All the Conclusion of the1 2 Hypotheses c Argument

p q �q p → �q q 1 ∧ 2 �p (1 ∧ 2) → c

T T

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1.

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4.


London—Devotees of writer LewisCarroll believe they have found what

inspired his grinning Cheshire Cat,made famous in his book “Alice’s Ad-ventures in Wonderland.”

Members of the Lewis Carroll Societymade the discovery over the weekend ina church at which the author’s father

was once rector in the Yorkshire villageof Croft in northern England.

It is a rough-hewn carving of a cat’shead smiling near an altar, probablydating to the 10th century. Seen frombelow and from the perspective of asmall boy, all that can be seen is thegrinning mouth.

Carroll’s Alice watched the CheshireCat disappear “ending with the grin,

Featured InThe news

Reprinted with permission fromReuters.

which remained for some time after therest of the head had gone.”

Alice mused: “I have often seen a catwithout a grin, but not a grin without acat. It is the most curious thing I haveseen in all my life.”

CHURCH CARVING MAY BE ORIGINAL‘CHESHIRE CAT’

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Hypothesis 2: Recopy the q column.

1 ∧ 2: A conjunction is true only when both components are true; enter a T in row 3and Fs elsewhere.

Conclusion c: A negation has the opposite truth values; enter an F in rows 1 and 2and a T in rows 3 and 4.

At this point, all that remains is the final column (see Figure 1.62).

54 CHAPTER 1 Logic

1.

2.

3.

4.

1 2 c

p q �q p → �q q 1 ∧ 2 �p (1 ∧ 2) → c

T T F F T F F

T F T T F F F

F T F T T T T

F F T T F F T

Truth values of the expressions.FIGURE 1.62

1 2 c

p q �q p → �q q 1 ∧ 2 �p (1 ∧ 2) → c

T T F F T F F T

T F T T F F F T

F T F T T T T T

F F T T F F T T

1.

2.

3.

4.

Truth table for the argument [(p → �q) ∧ q] → �p.FIGURE 1.63

The last column in the truth table is the conditional that represents the entireargument. A conditional is false only when its premise is true and its conclusion isfalse. The only instance in which the premise (1 ∧ 2) is true is row 3. Correspondingto this entry, the conclusion �p is also true. Consequently, the conditional (1 ∧ 2) → cis true in row 3. Because the premise (1 ∧ 2) is false in rows 1, 2, and 4, the condi-tional (1 ∧ 2) → c is automatically true in those rows as well. The completed truthtable is shown in Figure 1.63.

The completed truth table shows that the conditional [(p → �q) ∧ q] → �pis always true. The conditional represents the argument “If he is illiterate, hecannot fill out the application. He can fill out the application. Therefore, he is notilliterate.” Thus, the argument is valid.

Tautologies

A tautology is a statement that is always true. For example, the statement

“(a � b)2 � a2 � 2ab � b2 ”

is a tautology.

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EXAMPLE 2 DETERMINING WHETHER A STATEMENT IS A TAUTOLOGYDetermine whether the statement (p ∧ q) → (p ∨ q) is a tautology.

SOLUTION We need to construct a truth table for the statement. Because there are two letters,the table must have 22 � 4 rows. We need a column for (p ∧ q), one for (p ∨ q),and one for (p ∧ q) → (p ∨ q). The completed truth table is shown in Figure 1.64.


p q p ∧ q p ∨ q ( p ∧ q) → ( p ∨ q)

T T T T T

T F F T T

F T F T T

F F F F T

1.

2.

3.

4.

Truth table for the statement (p ∧ q) → (p ∨ q).FIGURE 1.64

Because (p ∧ q) → (p ∨ q) is always true, it is a tautology.

As we have seen, an argument can be represented by a single conditional. Ifthis conditional is always true, the argument is valid (and vice versa).

EXAMPLE 3 USING A TRUTH TABLE TO ANALYZE AN ARGUMENT Determinewhether the following argument is valid:“If the defendant is innocent, the defendant does not go to jail. The defendant doesnot go to jail. Therefore, the defendant is innocent.”

SOLUTION Separating the hypotheses from the conclusion, we have

1. If the defendant is innocent, the defendant does not go to jail.2. The defendant does not go to jail.Therefore, the defendant is innocent.

Now we define symbols to represent the various components of the statements:

p: The defendant is innocent.q: The defendant goes to jail.

Symbolically, the argument has the form

1. p → �q2. �q∴ p

and is represented by the conditional [(p → �q) ∧ �q] → p.

VALIDITY OF AN ARGUMENTAn argument having n hypotheses h1, h2, . . . , hn and conclusion c is valid ifand only if the conditional [h1 ∧ h2 ∧ . . . ∧ hn ] → c is a tautology.

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Now we construct a truth table with four rows, along with the necessarycolumns. The completed table is shown in Figure 1.65.

56 CHAPTER 1 Logic

Truth table for the argument [(p → �q) ∧ �q] → p.FIGURE 1.65

1.

2.

3.

4.

2 1 c

p q �q p → �q 1 ∧ 2 p (1 ∧ 2) → c

T T F F F T T

T F T T T T T

F T F T F F T

F F T T T F F

The column representing the argument has an F in row 4; therefore, the con-ditional representation of the argument is not a tautology. In particular, the conclu-sion does not logically follow the hypotheses when both p and q are false (row 4).The argument is not valid. Let us interpret the circumstances expressed in row 4,the row in which the argument breaks down. Both p and q are false—that is, the de-fendant is guilty and the defendant does not go to jail. Unfortunately, this situationcan occur in the real world; guilty people do not always go to jail! As long as it ispossible for a guilty person to avoid jail, the argument is invalid.

The following argument was presented as Example 6 in Section 1.1. In thatsection, we constructed a Venn diagram to show that the argument was in factvalid. We now show an alternative method; that is, we construct a truth table to de-termine whether the argument is valid.

EXAMPLE 4 USING A TRUTH TABLE TO ANALYZE AN ARGUMENT Determinewhether the following argument is valid: “No snake is warm-blooded. Allmammals are warm-blooded. Therefore, snakes are not mammals.”


1. No snake is warm-blooded.2. All mammals are warm-blooded.Therefore, snakes are not mammals.

These statements can be rephrased as follows:

1. If it is a snake, then it is not warm-blooded.2. If it is a mammal, then it is warm-blooded.Therefore, if it is a snake, then it is not a mammal.


p: It is a snake.q: It is warm-blooded.r: It is a mammal.

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1. p → �q2. r → q∴ p → �r

and is represented by the conditional [(p → �q) ∧ (r → q)] → (p → �r).Now we construct a truth table with eight rows (23 � 8), along with the nec-

essary columns. The completed table is shown in Figure 1.66.


1 2 c

p q r �q �r p → �q r → q 1 ∧ 2 p → �r (1 ∧ 2) → c

T T T F F F T F F T

T T F F T F T F T T

T F T T F T F F F T

T F F T T T T T T T

F T T F F T T T T T

F T F F T T T T T T

F F T T F T F F T T

F F F T T T T T T T

1.

2.

3.

4.

5.

6.

7.

8.

Truth table for the argument [(p → �q) ∧ (r → q)] → (p → �r).FIGURE 1.66

The preceding examples contained relatively simple arguments, each con-sisting of only two hypotheses and two simple statements (letters). In such cases,many people try to employ “common sense” to confirm the validity of the argu-ment. For instance, the argument “If it is raining, the streets are wet. It is raining.Therefore, the streets are wet” is obviously valid. However, it might not be so sim-ple to determine the validity of an argument that contains several hypotheses andmany simple statements. Indeed, in such cases, the argument’s truth table mightbecome quite lengthy, as in the next example.

EXAMPLE 5 USING A TRUTH TABLE TO ANALYZE AN ARGUMENT The followingwhimsical argument was written by Lewis Carroll and appeared in his 1896 bookSymbolic Logic:

“No ducks waltz. No officers ever decline to waltz. All my poultry are ducks. Therefore, my poultry are not officers.”

Construct a truth table to determine whether the argument is valid.

The last column of the truth table represents the argument and contains all T’s.Consequently, the conditional [(p → �q) ∧ (r → q)] → (p → �r) is a tautology;the argument is valid.

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1. No ducks waltz.2. No officers ever decline to waltz.3. All my poultry are ducks.Therefore, my poultry are not officers.

These statements can be rephrased as

1. If it is a duck, then it does not waltz.2. If it is an officer, then it does not decline to waltz.

(Equivalently, “If it is an officer, then it will waltz.”)3. If it is my poultry, then it is a duck.Therefore, if it is my poultry, then it is not an officer.

58 CHAPTER 1 Logic

CHARLES LUTWIDGE DODGSON, 1832–1898

To those who assume that it isimpossible for a person to

excel both in the creative worldsof art and literature and in the dis-ciplined worlds of mathematicsand logic, the life of CharlesLutwidge Dodgson is a wondrouscounterexample. Known the world overas Lewis Carroll, Dodgson penned thenonsensical classics Alice’s Adventures inWonderland and Through the LookingGlass. However, many people are sur-prised to learn that Dodgson (from ageeighteen to his death) was a permanentresident at the University at Oxford,teaching mathematics and logic. And asif that were not enough, Dodgson is nowrecognized as one of the leading portraitphotographers of the Victorian era.

The eldest son in a family of elevenchildren, Charles amused his youngersiblings with elaborate games, poems,stories, and humorous drawings. Thisattraction to entertaining children withfantastic stories manifested itself in muchof his later work as Lewis Carroll. Besideshis obvious interest in telling stories, the

young Dodgson wasalso intrigued bymathematics. At theage of eight, Charlesasked his father to ex-plain a book on log-arithms. When toldthat he was tooyoung to understand,Charles persisted,

“But please, explain!”The Dodgson family had a strong ec-

clesiastical tradition; Charles’s father,great-grandfather, and great-great-grand-father were all clergymen. Following inhis father’s footsteps, Charles attendedChrist Church, the largest and most cele-brated of all the Oxford colleges. Aftergraduating in 1854, Charles remained atOxford, accepting the position of mathe-matical lecturer in 1855. However, ap-pointment to this position was conditionalupon his taking Holy Orders in the Angli-can church and upon his remaining celi-bate. Dodgson complied and wasnamed a deacon in 1861.

The year 1856 was filled with eventsthat had lasting effects on Dodgson.Charles Lutwidge created his pseudonymby translating his first and middle names

into Latin (Carolus Ludovic), reversing theirorder (Ludovic Carolus), and translatingthem back into English (Lewis Carroll). Inthis same year, Dodgson began his“hobby” of photography. He is consid-ered by many to have been an artistic pi-oneer in this new field (photography wasinvented in 1839). Most of Dodgson’swork consists of portraits that chronicle theVictorian era, and over 700 photographstaken by Dodgson have been preserved.His favorite subjects were children, espe-cially young girls.

Dodgson’s affinity for children broughtabout a meeting in 1856 that wouldeventually establish his place in the his-tory of literature. Early in the year, Dodg-son met the four children of the dean ofChrist Church: Harry, Lorina, Edith, andAlice Liddell. He began seeing the chil-dren on a regular basis, amusing themwith stories and photographing them. Al-though he had a wonderful relationshipwith all four, Alice received his specialattention.

On July 4, 1862, while rowing andpicnicking with Alice and her sisters,Dodgson entertained the Liddell girls witha fantastic story of a little girl named Alicewho fell into a rabbit hole. Captivated by

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Young Alice Liddell inspired Lewis Carroll towrite Alice’s Adventures in Wonderland. Thisphoto is one of the many Carroll took of Alice.

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Carroll’s book The Game of Logic presents the studyof formalized logic in a gamelike fashion. After listingthe “rules of the game” (complete with gameboardand markers), Carroll captures the reader’s interestwith nonsensical syllogisms.

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the story, Alice Liddell insisted that Dodg-son write it down for her. He complied,initially titling it Alice’s Adventure Under-ground.

Dodgson’s friends subsequently encour-aged him to publish the manuscript, and in1865, after editing and inserting new

episodes, Lewis Carroll gave the worldAlice’s Adventures in Wonderland. Al-though the book appeared to be a whim-sical excursion into chaotic nonsense,Dodgson’s masterpiece contained manyexercises in logic and metaphor. The bookwas a success, and in 1871, a sequel,Through the Looking Glass, was printed.When asked to comment on the meaningof his writings, Dodgson replied, “I’m verymuch afraid I didn’t mean anything butnonsense! Still, you know, words meanmore than we mean to express when weuse them; so a whole book ought to meana great deal more than the writer means.So, whatever good meanings are in thebook, I’m glad to accept as the meaningof the book.”

In addition to writing “children’s sto-ries,” Dodgson wrote numerous mathe-matics essays and texts, including TheFifth Book of Euclid Proved Alge-braically, Formulae of Plane Trigonome-try, A Guide to the Mathematical Stu-dent, and Euclid and His Modern Rivals.In the field of formal logic, Dodgson’sbooks The Game of Logic (1887) andSymbolic Logic (1896) are still used assources of inspiration in numerousschools worldwide.


p: It is a duck.q: It will waltz.r: It is an officer.s: It is my poultry.


1. p → �q2. r → q3. s → p∴ s → �r

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1.5 Exercises

60 CHAPTER 1 Logic

In Exercises 1–10, use the given symbols to rewrite the argumentin symbolic form.

1. p: It is raining.q: The streets are wet. } Use these symbols.

1. If it is raining, then the streets are wet.2. It is raining.

Therefore, the streets are wet.

2. p: I have a college degree.q: I am lazy. } Use these symbols.

1. If I have a college degree, I am not lazy.2. I do not have a college degree.

Therefore, I am lazy.

3. p: It is Tuesday.

q: The tour group is in Belgium. } Use these symbols.

1. If it is Tuesday, then the tour group is in Belgium.

2. The tour group is not in Belgium.

Therefore, it is not Tuesday.

4. p: You are a gambler.

q: You have financial security. } Use these symbols.

1. You do not have financial security if you are agambler.

2. You do not have financial security.

Therefore, you are a gambler.

1 2 3 c

p q r s �q �r p → �q r → q s → p 1 ∧ 2 ∧ 3 s → �r (1 ∧ 2 ∧ 3) → c

T T T T F F F T T F F T

T T T F F F F T T F T T

T T F T F T F T T F T T

T T F F F T F T T F T T

T F T T T F T F T F F T

T F T F T F T F T F T T

T F F T T T T T T T T T

T F F F T T T T T T T T

F T T T F F T T F F F T

F T T F F F T T T T T T

F T F T F T T T F F T T

F T F F F T T T T T T T

F F T T T F T F F F F T

F F T F T F T F T F T T

F F F T T T T T F F T T

F F F F T T T T T T T T

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

Truth table for the argument [(p → �q) ∧ (r → q) ∧ (s → p)] → (s → �r).FIGURE 1.67

Now we construct a truth table with sixteen rows (24 � 16), along with the neces-sary columns. The completed table is shown in Figure 1.67.

The last column of the truth table represents the argument and contains allT’s. Consequently, the conditional [(p → �q) ∧ (r → q) ∧ (s → p)] → (s → �r) isa tautology; the argument is valid.

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17. the argument in Exercise 718. the argument in Exercise 819. the argument in Exercise 920. the argument in Exercise 10

In Exercises 21–42, define the necessary symbols, rewritethe argument in symbolic form, and use a truth table todetermine whether the argument is valid. If the argument isinvalid, interpret the specific circumstances that cause theargument to be invalid.

21. 1. If the Democrats have a majority, Smith is appointed and student loans are funded.

2. Smith is appointed or student loans are not funded.

Therefore, the Democrats do not have a majority.

22. 1. If you watch television, you do not read books.2. If you read books, you are wise.

Therefore, you are not wise if you watch television.

23. 1. If you argue with a police officer, you get a ticket.2. If you do not break the speed limit, you do not get a

ticket.

Therefore, if you break the speed limit, you argue witha police officer.

24. 1. If you do not recycle newspapers, you are not anenvironmentalist.

2. If you recycle newspapers, you save trees.

Therefore, you are an environmentalist only if yousave trees.

25. 1. All pesticides are harmful to the environment.2. No fertilizer is a pesticide.

Therefore, no fertilizer is harmful to the environment.

26. 1. No one who can afford health insurance is unem-ployed.

2. All politicians can afford health insurance.

Therefore, no politician is unemployed.

27. 1. All poets are loners.2. All loners are taxi drivers.

Therefore, all poets are taxi drivers.

28. 1. All forest rangers are environmentalists.2. All forest rangers are storytellers.

Therefore, all environmentalists are storytellers.

29. 1. No professor is a millionaire.2. No millionaire is illiterate.

Therefore, no professor is illiterate.

30. 1. No artist is a lawyer.

2. No lawyer is a musician.

Therefore, no artist is a musician.

31. 1. All lawyers study logic.2. You study logic only if you are a scholar.3. You are not a scholar.

Therefore, you are not a lawyer.

5. p: You exercise regularly.

q: You are healthy. } Use these symbols.

1. You exercise regularly only if you are healthy.

2. You do not exercise regularly.

Therefore, you are not healthy.

6. p: The senator supports new taxes.

q: The senator is reelected. }1. The senator is not reelected if she supports new taxes.

2. The senator does not support new taxes.

Therefore, the senator is reelected.

7. p: A person knows Morse code.q: A person operates a telegraph. }r: A person is Nikola Tesla.


2. Nikola Tesla knows Morse code.

Therefore, Nikola Tesla operates a telegraph.

HINT: Hypothesis 2 can be symbolized as r ∧ p.

8. p: A person knows CPR.q: A person is a paramedic.r: A person is David Lee Roth.


2. David Lee Roth is a paramedic.

Therefore, David Lee Roth knows CPR.

HINT: Hypothesis 2 can be symbolized as r ∧ q.

9. p: It is a monkey.q: It is an ape. } Use these symbols.r: It is King Kong.


2. King Kong is an ape.

Therefore, King Kong is not a monkey.

10. p: It is warm-blooded.q: It is a reptile. } Use these symbols.r: It is Godzilla.

1. Being warm-blooded is sufficient for not being areptile.

2. Godzilla is not warm-blooded.

Therefore, Godzilla is a reptile.

In Exercises 11–20, use a truth table to determine the validity ofthe argument specified. If the argument is invalid, interpret thespecific circumstances that cause it to be invalid.

11. the argument in Exercise 1






Use thesesymbols.}

Use thesesymbols.

Use thesesymbols.

1.5 Exercises 61

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of Hypothesis 2 can be symbolized as

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a police officer.

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1. If you watch television, you do not read books.

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1. If you watch television, you do not read books.

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2. If you read books, you are wise.

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2. If you read books, you are wise.


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1. If you argue with a police officer, you get a ticket.

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32. 1. All licensed drivers have insurance.

2. You obey the law if you have insurance.

3. You obey the law.

Therefore, you are a licensed driver.

33. 1. Drinking espresso is sufficient for not sleeping.


3. Not eating dessert is sufficient for drinking espresso.

Therefore, not being on a diet is necessary for sleeping.

34. 1. Not being eligible to vote is sufficient for ignoringpolitics.

2. Not being a convicted felon is necessary for beingeligible to vote.

3. Ignoring politics is sufficient for being naive.

Therefore, being naive is necessary being a convictedfelon.

35. If the defendant is innocent, he does not go to jail. Thedefendant goes to jail. Therefore, the defendant is guilty.

36. If the defendant is innocent, he does not go to jail. Thedefendant is guilty. Therefore, the defendant goes to jail.

37. If you are not in a hurry, you eat at Lulu’s Diner. If youare in a hurry, you do not eat good food. You eat atLulu’s. Therefore, you eat good food.

38. If you give me a hamburger today, I pay you tomorrow.If you are a sensitive person, you give me a hamburgertoday. You are not a sensitive person. Therefore, I donot pay you tomorrow.

39. If you listen to rock and roll, you do not go to heaven. Ifyou are a moral person, you go to heaven. Therefore,you are not a moral person if you listen to rock and roll.

40. If you follow the rules, you have no trouble. If you arenot clever, you have trouble. You are clever. Therefore,you do not follow the rules.

41. The water not being cold is sufficient for going swim-ming. Having goggles is necessary for going swim-ming. I have no goggles. Therefore, the water is cold.

42. I wash my car only if the sky is clear. The sky not beingclear is necessary for it to rain. I do not wash my car.Therefore, it is raining.

The arguments given in Exercises 43–50 were written by LewisCarroll and appeared in his 1896 book Symbolic Logic. For eachargument, define the necessary symbols, rewrite the argument insymbolic form, and use a truth table to determine whether theargument is valid.

43. 1. All medicine is nasty.2. Senna is a medicine.

Therefore, senna is nasty.

NOTE: Senna is a laxative extracted from the driedleaves of cassia plants.

44. 1. All pigs are fat.

2. Nothing that is fed on barley-water is fat.

Therefore, pigs are not fed on barley-water.

62 CHAPTER 1 Logic

45. 1. Nothing intelligible ever puzzles me.

2. Logic puzzles me.

Therefore, logic is unintelligible.

46. 1. No misers are unselfish.

2. None but misers save eggshells.

Therefore, no unselfish people save eggshells.

47. 1. No Frenchmen like plum pudding.

2. All Englishmen like plum pudding.

Therefore, Englishmen are not Frenchmen.

48. 1. A prudent man shuns hyenas.

2. No banker is imprudent.

Therefore, no banker fails to shun hyenas.

49. 1. All wasps are unfriendly.

2. No puppies are unfriendly.

Therefore, puppies are not wasps.

50. 1. Improbable stories are not easily believed.

2. None of his stories are probable.

Therefore, none of his stories are easily believed.



51. What is a tautology?

52. What is the conditional representation of an argument?

53. Find a “logical” argument in a newspaper article, anadvertisement, or elsewhere in the media. Analyze thatargument and discuss the implications.


54. What was Charles Dodgson’s pseudonym? How did heget it? What classic “children’s stories” did he write?

55. What did Charles Dodgson contribute to the study offormal logic?

56. Charles Dodgson was a pioneer in what artistic field?

57. Who was Alice Liddell?

Web project

58. Write a research paper on any historical topic referredto in this chapter or a related topic. Below is a partiallist of topics.● Aristotle● George Boole● Augustus De Morgan● Charles Dodgson/Lewis Carroll● Gottfried Wilhelm LeibnizSome useful links for this web project are listed on thetext web site: www.cengage.com/math/johnson

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Concept Questions

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1. All wasps are unfriendly.

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1. Improbable stories are not easily believed.

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Therefore, none of his stories are easily believed.

8. 1. Some animals are dangerous.

2. A gun is not an animal.

Therefore, a gun is not dangerous.

9. 1. Some contractors are electricians.

2. All contractors are carpenters.

Therefore, some electricians are carpenters.

10. Solve the following sudoku puzzle.

11. Explain why each of the following is or is not astatement.

a. The Golden Gate Bridge spans Chesapeake Bay.

b. The capital of Delaware is Dover.

c. Where are you spending your vacation?

d. Hawaii is the best place to spend a vacation.


a. All of the lawyers are ethical.

b. Some of the lawyers are ethical.

c. None of the lawyers is ethical.

d. Some of the lawyers are not ethical.

4

1

6

8

5

2

3

2

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3

7

4

6

5

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REVIEW EXERCISES

1TERMSargumentbiconditionalcompound statementconclusionconditionalconjunctioncontrapositive

conversedeductive reasoningdisjunctionequivalent expressionsexclusive orhypothesisimplicationinclusive orinductive reasoning

invalid argumentinverselogicnecessarynegationpremisequantifierreasoningstatement

sudoku sufficientsyllogismtautologytruth tabletruth valuevalid argumentVenn diagram

1. Classify each argument as deductive or inductive.a. 1. Hitchcock’s “Psycho” is a suspenseful movie.

2. Hitchcock’s “The Birds” is a suspenseful movie.

Therefore, all Hitchcock movies are suspenseful.b. 1. All Hitchcock movies are suspenseful.

2. “Psycho” is a Hitchcock movie.

Therefore, “Psycho” is suspenseful.2. Explain the general rule or pattern used to assign the

given letter to the given word. Fill in the blank with theletter that fits the pattern.

3. Fill in the blank with what is most likely to be the nextnumber. Explain the pattern generated by your answer.

1, 6, 11, 4, _____

In Exercises 4–9, construct a Venn diagram to determine thevalidity of the given argument.

4. 1. All truck drivers are union members.

2. Rocky is a truck driver.

Therefore, Rocky is a union member.

5. 1. All truck drivers are union members.

2. Rocky is not a truck driver.

Therefore, Rocky is not a union member.

6. 1. All mechanics are engineers.

2. Casey Jones is an engineer.

Therefore, Casey Jones is a mechanic.

7. 1. All mechanics are engineers.

2. Casey Jones is not an engineer.

Therefore, Casey Jones is not a mechanic.

day morning afternoon dusk night

y r f s _____

Chapter Review

63

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1. Some animals are dangerous.

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9.

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9. 1. Some contractors are electricians.

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1. Some contractors are electricians.Therefore, all Hitchcock movies are suspenseful.

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In Exercises 18–25, construct a truth table for the compoundstatement.

18. p ∨ �q 19. p ∧ �q

20. �p → q 21. (p ∧ q) → �q

22. q ∨ �(p ∨ r) 23. �p → (q ∨ r)

24. (q ∧ p) → (�r ∨ p) 25. (p ∨ r) → (q ∧ �r)

In Exercises 26–30, construct a truth table to determine whetherthe statements in each pair are equivalent.

26. The car is unreliable or expensive.If the car is reliable, then it is expensive.

27. If I get a raise, I will buy a new car.If I do not get a raise, I will not buy a new car.

28. She is a Democrat or she did not vote.She is not a Democrat and she did vote.

29. The raise is not unjustified and the managementopposes it.It is not the case that the raise is unjustified or themanagement does not oppose it.

30. Walking on the beach is sufficient for not wearingshoes. Wearing shoes is necessary for not walking onthe beach.

In Exercises 31–38, write a sentence that represents the negationof each statement.

31. Jesse had a party and nobody came.

32. You do not go to jail if you pay the fine.

33. I am the winner or you are blind.

34. He is unemployed and he did not apply for financialassistance.

35. The selection procedure has been violated if hisapplication is ignored.

36. The jackpot is at least $1 million.

37. Drinking espresso is sufficient for not sleeping.


39. Given the statements

p: You are an avid jogger.

q: You are healthy.

write the sentence represented by each of thefollowing.

a. p → q b. q → p

c. �p → �q d. �q → �p

e. p ↔ q

40. Form (a) the inverse, (b) the converse, and (c) thecontrapositive of the conditional “If he is elected, thecountry is in big trouble.”

In Exercises 41 and 42, express the contrapositive of the givenconditional in terms of (a) a sufficient condition, and (b) anecessary condition.

41. Having a map is sufficient for not being lost.

42. Having syrup is necessary for eating pancakes.


a. His car is not new.

b. Some buildings are earthquake proof.

c. All children eat candy.

d. I never cry in a movie theater.


p: The television program is educational.

q: The television program is controversial.

express the following compound statements in symbolicform.a. The television program is educational and contro-

versial.b. If the television program isn’t controversial, it isn’t

educational.c. The television program is educational and it isn’t

controversial.d. The television program isn’t educational or

controversial.e. Not being controversial is necessary for a television

program to be educational.f. Being controversial is sufficient for a television

program not to be educational.


p: The advertisement is effective.q: The advertisement is misleading.r: The advertisement is outdated.

express the following compound statements in symbolicform.

a. All misleading advertisements are effective.b. It is a current, honest, effective advertisement.c. If an advertisement is outdated, it isn’t effective.d. The advertisement is effective and it isn’t misleading

or outdated.e. Not being outdated or misleading is necessary for

an advertisement to be effective.f. Being outdated and misleading is sufficient for an

advertisement not to be effective.


p: It is expensive.

q: It is undesirable.

express the following in words.

a. p → �q b. q ↔ �pc. �(p ∨ q) d. (p ∧ �q) ∨ (�p ∧ q)


p: The movie is critically acclaimed.q: The movie is a box office hit.r: The movie is available on DVD.

express the following in words.

a. (p ∨ q) → r b. (p ∧ �q) → �rc. �(p ∨ q) ∧ r d. �r → (�p ∧ �q)

64 CHAPTER 1 Logic

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the beach.

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of each statement.

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32. You do not go to jail if you pay the fine.

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She is a Democrat or she did not vote.

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She is a Democrat or she did not vote.She is not a Democrat and she did vote.

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She is not a Democrat and she did vote.

The raise is not unjustified and the management

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The raise is not unjustified and the management

It is not the case that the raise is unjustified or the

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56. 1. Practicing is sufficient for making no mistakes.

2. Making a mistake is necessary for not receiving anaward.

3. You receive an award.

Therefore, you practice.

57. 1. Practicing is sufficient for making no mistakes.

2. Making a mistake is necessary for not receiving anaward.

3. You do not receive an award.

Therefore, you do not practice.

In Exercises 58–66, define the necessary symbols, rewrite theargument in symbolic form, and use a truth table to determinewhether the argument is valid.

58. If the defendant is guilty, he goes to jail. The defendantdoes not go to jail. Therefore, the defendant is notguilty.

59. I will go to the concert only if you buy me a ticket.You bought me a ticket. Therefore, I will go to theconcert.

60. If tuition is raised, students take out loans or drop out.If students do not take out loans, they drop out.Students do drop out. Therefore, tuition is raised.

61. If our oil supply is cut off, our economy collapses. If wego to war, our economy doesn’t collapse. Therefore, ifour oil supply isn’t cut off, we do not go to war.

62. No professor is uneducated. No monkey is educated.Therefore, no professor is a monkey.

63. No professor is uneducated. No monkey is a professor.Therefore, no monkey is educated.

64. Vehicles stop if the traffic light is red. There is noaccident if vehicles stop. There is an accident.Therefore, the traffic light is not red.

65. Not investing money in the stock market is necessaryfor invested money to be guaranteed. Invested moneynot being guaranteed is sufficient for not retiring at anearly age. Therefore, you retire at an early age only ifyour money is not invested in the stock market.

66. Not investing money in the stock market is necessaryfor invested money to be guaranteed. Invested moneynot being guaranteed is sufficient for not retiring at anearly age. You do not invest in the stock market.Therefore, you retire at an early age.

Determine the validity of the arguments in Exercises 67 and 68 byconstructing a

a. Venn diagram and a

b. truth table.

c. How do the answers to parts (a) and (b) compare? Why?

67. 1. If you own a hybrid vehicle, then you are anenvironmentalist.

2. You are not an environmentalist.

Therefore, you do not own a hybrid vehicle.

In Exercises 43–48, (a) determine the premise and conclusionand (b) rewrite the compound statement in the standard “if . . .then . . .” form.

43. The economy improves only if unemployment goesdown.

44. The economy improves if unemployment goes down.

45. No computer is unrepairable.

46. All gemstones are valuable.

47. Being the fourth Thursday in November is sufficientfor the U.S. Post Office to be closed.

48. Having diesel fuel is necessary for the vehicle tooperate.

In Exercises 49 and 50, translate the two statements into symbolicform and use truth tables to determine whether the statements areequivalent.

49. If you are allergic to dairy products, you cannot eatcheese.

If you cannot eat cheese, then you are allergic to dairyproducts.

50. You are a fool if you listen to me.

You are not a fool only if you do not listen to me.

51. Which pairs of statements are equivalent?

i. If it is not raining, I ride my bicycle to work.

ii. If I ride my bicycle to work, it is not raining.

iii. If I do not ride my bicycle to work, it is raining.

iv. If it is raining, I do not ride my bicycle to work.

In Exercises 52–57, define the necessary symbols, rewrite theargument in symbolic form, and use a truth table to determinewhether the argument is valid.

52. 1. If you do not make your loan payment, your car isrepossessed.

2. Your car is repossessed.

Therefore, you did not make your loan payment.

53. 1. If you do not pay attention, you do not learn thenew method.

2. You do learn the new method.

Therefore, you do pay attention.

54. 1. If you rent DVD, you will not go to the movietheater.

2. If you go to the movie theater, you pay attention tothe movie.

Therefore, you do not pay attention to the movie if yourent DVDs.

55. 1. If the Republicans have a majority, Farnsworth isappointed and no new taxes are imposed.

2. New taxes are imposed.

Therefore, the Republicans do not have a majority orFarnsworth is not appointed.

Chapter Review 65

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Propert

y argument in symbolic form, and use a truth table to determine

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y argument in symbolic form, and use a truth table to determine

1. If you do not make your loan payment, your car is

Propert

y 1. If you do not make your loan payment, your car is

Propert

y 2. Your car is repossessed.

Propert

y 2. Your car is repossessed.

Therefore, you did not make your loan payment.

Propert

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new method.

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2. You do learn the new method.

of In Exercises 52–57, define the necessary symbols, rewrite theof In Exercises 52–57, define the necessary symbols, rewrite theargument in symbolic form, and use a truth table to determineof argument in symbolic form, and use a truth table to determine

Cenga

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Cenga

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60.

Cenga

ge 60. If tuition is raised, students take out loans or drop out.

Cenga

ge If tuition is raised, students take out loans or drop out.

If students do not take out loans, they drop out.

Cenga

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If students do not take out loans, they drop out.Students do drop out. Therefore, tuition is raised.

Cenga

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Students do drop out. Therefore, tuition is raised.

61.

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61.

If I ride my bicycle to work, it is not raining.

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If I ride my bicycle to work, it is not raining.

If I do not ride my bicycle to work, it is raining.

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If I do not ride my bicycle to work, it is raining.

If it is raining, I do not ride my bicycle to work.Cenga

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If it is raining, I do not ride my bicycle to work.

In Exercises 52–57, define the necessary symbols, rewrite theCen

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In Exercises 52–57, define the necessary symbols, rewrite the

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argument in symbolic form, and use a truth table to determine

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argument in symbolic form, and use a truth table to determinewhether the argument is valid.

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whether the argument is valid.

If the defendant is guilty, he goes to jail. The defendant

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If the defendant is guilty, he goes to jail. The defendantdoes not go to jail. Therefore, the defendant is not

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does not go to jail. Therefore, the defendant is not

I will go to the concert only if you buy me a ticket.

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I will go to the concert only if you buy me a ticket.You bought me a ticket. Therefore, I will go to theLe

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concert.

71. a. What is a sufficient condition?

b. What is a necessary condition?

72. What is a tautology?

73. When constructing a truth table, how do you determinehow many rows to create?


74. What role did the following people play in thedevelopment of formalized logic?● Aristotle● George Boole● Augustus De Morgan● Charles Dodgson● Gottfried Wilhelm Leibniz

68. 1. If you own a hybrid vehicle, then you are anenvironmentalist.

2. You are an environmentalist.

Therefore, you own a hybrid vehicle.



69. What is a statement?

70. a. What is a disjunction? Under what conditions is adisjunction true?

b. What is a conjunction? Under what conditions is aconjunction true?

c. What is a conditional? Under what conditions is aconditional true?

d. What is a negation? Under what conditions is anegation true?

66 CHAPTER 1 Logic

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Gottfried Wilhelm Leibniz

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Gottfried Wilhelm Leibniz

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95057 01 ch01 p001-066 - cengage.com 1 Logic.pdf · much an exercise in logic as it is a fantasy ... 24 4 2 x 12 2 ; 212 22 142111 2 2112 ... The validity of a deductive argument