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The Foundation: Logic Conditional Statements
Muhammad Ariefdownload dari http://arief.ismy.web.id
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• Discrete Mathematics and its Applications
Kenneth H. Rosen – 6th edition
Mc Graw-Hill International Edition
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Conditional Statements• “If p then q”
“p implies q” “p q”; p: hypothesis, q: conclusion.
• Conditional: the truth of statement q is conditioned on the truth of statement p
• Example:IF 36 is divisible by 6, THEN 36 is divisible by 3
• IF you show up for work Monday morning, THEN you will get the job.
• Under what circumstances is the above sentence false?
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Truth Table for Conditional Statements
Definition
• p q is false when p is true and q is false; otherwise it is true.
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Truth Table for p ~q ~p
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p q r (p r) (q r)OR p q r (p r) (q r)
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Representation of IF-THEN as OR
• p: you do not get to work on time
• q: you are fired
• IF you do not get to work on time THEN you are fired
• ~p: you get to work on time
• p q ~p q
• You get to work on time OR you are fired
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Representation of IF-THEN as OR
• Truth table for p q ~p q
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De Morgan’s LawsDefinition:• The negation of an AND statement is
logically equivalent to the OR statement in which each component is negated.
~(p q) ~p ~q
• The negation of an OR statement is logically equivalent to the AND statement in which each component is negated.
~(p q) ~p ~qhttp://arief.ismy.web.id
The Negation of a Conditional Statement
• The negation of “IF p THEN q” is logically equivalent to “p and not q”
~(p q) p ~q• Show the equivalence by using Morgan
Law• ~(p q) ~(~p q) ~(~p) ~q p ~q
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The Negation of a Conditional Statement
• ~(IF my car is in the repair shop, THEN I cannot get the class)
• My car is in the repair shop and I can get to class
• ~(IF Sara lives in Athens, THEN she lives in Greece)
• Sara lives in Athens and she does not live in Greece
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The Contrapositive of a Conditional Statements
Definition• The contrapositive of a conditional statement of
the form “IF p THEN q” is “IF ~q THEN ~p”• The contrapositive of p q is ~q ~p
• A conditional statement is logically equivalent to its contrapositive.
Construct the truth table
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Example
• IF Howard can swim across the lake, THEN Howard can swim to the island
• IF Howard cannot swim to the island, then Howard cannot swim across the lake
• IF today is Easter, THEN tomorrow is Monday• IF tomorrow is not Monday, THEN today is not
Easter
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The Converse and Inverse of a Conditional Statement
Definition• Suppose a conditional statement of the form “IF p
THEN q” is given.• The converse is “IF q THEN p”• The inverse is “IF ~p THEN ~q”
Symbolically• The converse of p q is q p• The inverse of p q is ~p ~q
Are they logically equivalent??
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p q ~p ~q p q q p ~p ~q
T T F F T T T
T F F T F T T
F T T F T F F
F F T T T T T
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Example: Converse, Inverse
• IF Howard can swim across the lake, THEN Howard can swim to the island
• Converse: IF Howard can swim to the island THEN Howard can swim across the lake
• Inverse: IF Howard cannot swim across the lake, THEN Howard cannot swim to the island.
• IF today is Easter, THEN tomorrow is Monday• Converse: IF tomorrow is Monday, THEN today
is Easter• Inverse: IF today is not Easter, THEN tomorrow
is not Mondayhttp://arief.ismy.web.id
Only IfDefinition• p and q are statements,• p only if q means “IF not q THEN not p”• Or, equivalently,• “IF p THEN q”
• John will break the world’s record for the mile run ONLY IF he runs the mile in under four minutes.
• IF John does not run the mile in under four minutes, THEN he will not break the world’s record
• IF John breaks the world’s record, THEN he will have run the mile in under four minutes
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Biconditional
Definition• Given statement variables p and q, the biconditional of p
and q is “ p if, and only if, q” and is denoted p q. The words if and only if are sometimes abbreviated iff.
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Biconditional
• Is “ p if, and only if, q” logically equivalent with “ p if q “ and “ p only if q” ?
p q (q p) (p q)
Construct the truth table
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p q (q p) (p q)
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Necessary and Sufficient Conditions
Definition• If r and s are statements:
– r is a sufficient condition for s means “if r then s”
– r is a necessary condition for s means “if not r then not s” or “if s then r”
• r is a necessary and sufficient condition for s means “r if, and only if, s”.
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