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
ARITHMETIC
ALGEBRA
Algebra is a generalization of Arithmetic.
Mathematics Division, IMSP, UPLB
ALGEBRA AND ARITHMETIC
ASSIGNMENT:
1.
2.
What is the Fundamental
Theorem of Arithmetic?
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
Examples (you can also use < >)
[Z2, +2] (binary arithmetic)
[Z4, x4]
[Z12, +12] (clock arithmetic)
[DC, “followed by”]
[R, +, · ]
Mathematics Division, IMSP, UPLB
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
SETS
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
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
belongs to set A.
Otherwise, we write
a  A, in case a
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
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
{ } or .
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
{x|x is a letter from the English alphabet}
A, E, I,
O, U
Q, W, R,
T, Y, P, S,
D, F, G, H,
J, K, L
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
1. {1,2,3}?
2. {1,2,3,4,5}?
3. {1,2,3,4,5,6}?