Ang aking kontrata:

Ako, si ______________, ay nangangakong magsisipag mag-aral hindi lang para sa aking sarili kundi para rin sa aking pamilya, para sa aking bayang Pilipinas at para sa ikauunlad ng mundo.

MATH 17: College Algebra and Trigonometry

Lecturer: Jomar Fajardo Rabajante 1st Semester AY 2012-2013 Math Division, IMSP, UPLB

SOME PRINCIPLES OF LEARNING

• Law of Use and Disuse

• Law of Frequency and Recency

• Law of Intensity

• Law of Readiness / Law of Mindset

• Law of Belonging and Motivation

Mathematics Division, IMSP, UPLB

MATHEMATICS

MATHs (Sipnayan)

Math is the study of patterns

Math is a language

Math is a tool for solving problems

Mathematics Division, IMSP, UPLB

ALGEBRA

ETYMOLOGY OF “ALGEBRA”

From Arabic word “al-jabr” which literally means “restoration”

Mathematics Division, IMSP, UPLB

ALGEBRA AND ARITHMETIC

Mathematics Division, IMSP, UPLB

Algebra is a generalization of Arithmetic.

ARITHMETIC ALGEBRA

ALGEBRA AND ARITHMETIC

ASSIGNMENT:

1. What is the Fundamental Theorem of Arithmetic?

2. What is the Fundamental Theorem of Algebra?

Mathematics Division, IMSP, UPLB

ALGEBRA

DISCIPLINES UNDER ALGEBRA

Elementary Algebra (e.g. HS and College Algebra)

Linear Algebra

Modern (Abstract) Algebra

Etc… Mathematics Division, IMSP, UPLB

ABSTRACTION OF ALGEBRA

ALGEBRA as the study of STRUCTURES

- study of the structure of abstract mathematical systems

Mathematics Division, IMSP, UPLB

ABSTRACTION OF ALGEBRA

Mathematics Division, IMSP, UPLB

Examples (you can also use < >)

[Z2, +2] (binary arithmetic)

[Z4, x4]

[Z12, +12] (clock arithmetic)

[DC, “followed by”]

[R, +, · ]

ABSTRACTION OF ALGEBRA

Algebraic Structure/Algebraic System

An algebraic structure is an abstract mathematical system involving

1) sets of objects and

2) some operations on those objects,

usually satisfying certain axioms.

Mathematics Division, IMSP, UPLB

ABSTRACTION OF ALGEBRA

Mathematics Division, IMSP, UPLB

SETS (the basic ingredient

of algebra)

Mathematics Division, IMSP, UPLB

A SET is a well-defined collection of objects, real or imagined.

Examples: Which of the following is a set?

1. Collection of all cities in the Philippines

2. Collection of good character actresses in Philippine movies

3. Collection of beautiful flowers in the Botanical Garden

4. Collection of all small positive numbers

5. Collection of all letters in the Filipino alphabet

Mathematics Division, IMSP, UPLB

SETS

SETS

WHY WELL-DEFINED?

For us to be able to determine/describe the elements of a set without confusion!

For us to be able to determine whether a given object belongs to the set or not.

Mathematics Division, IMSP, UPLB

SETS

Do we really need to know the notion of sets?

• Consider you are a bachelor looking for a girlfriend…

• Consider you are a grade 1 student…

Mathematics Division, IMSP, UPLB

SETS

• To name a set, we usually use capital letters.

• The objects that make up a set are called elements.

• In symbols, we write a A, in case a

belongs to set A.

Otherwise, we write a A.

Mathematics Division, IMSP, UPLB

METHODS FOR DESCRIBING A SET

• roster or enumeration method

• rule or set builder method

Mathematics Division, IMSP, UPLB

METHODS FOR DESCRIBING A SET

Consider the set of all letters in the word “baybayin”. Let us name it as set W.

Roster or enumeration method:

W={b,a,y,i,n}

Mathematics Division, IMSP, UPLB

METHODS FOR DESCRIBING A SET

Consider the set of all letters in the word “baybayin”. Let us name it as set W.

Roster or enumeration method:

W={b,a,y,i,n}

Mathematics Division, IMSP, UPLB

METHODS FOR DESCRIBING A SET

Consider the set of all letters in the word “baybayin”. Let us name it as set W.

Rule or set builder method:

W={x | x is a letter in the word “baybayin”}

Mathematics Division, IMSP, UPLB

METHODS FOR DESCRIBING A SET

Consider the set of all letters in the word “baybayin”. Let us name it as set W.

Rule or set builder method:

W={x | x is a letter in the word “baybayin”}

Mathematics Division, IMSP, UPLB

METHODS FOR DESCRIBING A SET

Consider the set of all letters in the word “baybayin”. Let us name it as set W.

Rule or set builder method:

W={x | x is a letter in the word “baybayin”}

“such that”

METHODS FOR DESCRIBING A SET

Consider the set of all letters in the word “baybayin”. Let us name it as set W.

Rule or set builder method:

W={x | x is a letter in the word “baybayin”}

Variable that represents any element in the set

METHODS FOR DESCRIBING A SET

Consider the set of all letters in the word “baybayin”. Let us name it as set W.

Rule or set builder method:

W={x | x is a letter in the word “baybayin”}

Criteria for deciding whether an object belongs to the set

EXAMPLE

• Set of all possible outcomes when you throw a coin

• Set of all undergraduate subjects offered in UPLB

Questions:

When is the rule method more useful than the roster method?

When is the roster method more useful than the rule method?

EXAMPLE

Can you write D using the rule method? D={butter, bread, coffee, rice}

Can you write P using the roster method? P={x | x is a point in a line}

EMPTY SET

There are sets that are so special they are given specific names.

One of them is the empty set, denoted by { } or .

The empty set has no element.

Examples are:

1. The set of snakes that can fly

2. The set of dogs that can talk

Mathematics Division, IMSP, UPLB

TRIVIA

Is this an empty set?

{}

Mathematics Division, IMSP, UPLB

UNIVERSAL SET

Another special set is the universal set denoted by U.

The universal set is a set that contains all objects under consideration.

UNIVERSAL SET

Q, W, R, T, Y, P, S,

D, F, G, H, J, K, L

{x|x is a letter from the English alphabet}

A, E, I, O, U

EXAMPLES

Give a universal set from which the members of the following sets could be chosen:

1. {4, 8, 12, 16}

2. {Rizal, Bonifacio, Mabini}

QUESTIONS

Can U={1,2,3,4,5} be a universal set of

2. {1,2,3,4,5}?

1. {1,2,3}?

3. {1,2,3,4,5,6}?