A. Write SET if the statement is a set,
otherwise write NOT SET
1.
2.
3.
Collection of planets in the solar system
Collection of provinces near Laguna
Collection of intelligent students
B. Given: S = set of odd numbers between 1
and 20 that is divisible b
EXERCISES 1.4
In each of the following, a) find the sum of the
polynomials, and b) subtract the second from the
first.
1.
2.
3.
4.
In each of the following, remove the symbols of
grouping and combine similar terms.
5.
6.
7.
8.
In each of the following pol
EXERCISES 1.5
Perform the indicated operations and simplify.
Final answers should be in simplest form.
1.
2.
3.
4.
5.
6.
7.
x 2 xy
3x 2 x3
24 x 2
6x2 9x
a 2 4a 4
a2 4
x 2 3x 4
x 2 x 12
2a 2 5a 12
4a 2 4a 3
xa xb ya yb
xm ym xn yn
( 4a 2 9b 2 )( 18a 12 )
(
EXERCISES 1.3
In problems 1 through 10, carry out the indicated operations and express your answer in the
form
.
1. a)
b)
2. a)
b)
3. a)
b)
4. a)
b)
c)
d)
5. a)
b)
6. a)
1
a bi
7. a)
b)
1
a bi
b)
8.
9.
10.
11. Show that the sum of two pure imaginaries
Chapter 1.2
Arguments
and Reasoning
1
Introduction to Reasoning and
Logic
Every good mathematician is
at least half a philosopher and
every good philosopher is at
least half a mathematician.
-FLG Frege
3
There is a need to understand the general principle
PART 3
THINKING
MATHEMATICALL
Y
3.1 MATHEMATICS
AS AN AXIOMATICDEDUCTIVE SYSTEM
ALGEBRA: The Language of Mathematics
Algebra may be described as a
generalization and extension of arithmetic.
Arithmetic is concerned primarily with the
effect of certain o
MATH 1
QUANTITATIVE
REASONING
1
WHY TAKE THIS COURSE?
Quantitative concepts and
skills are crucial to meeting
the difficult challenges in this
rapidly changing
technological world.
The ability to reason with
quantitative information is an
essential compon
CHAPTER 1.3
THE COMPLEX
NUMBER SYSTEM
Solvable Equations in R
ax b 0 is always solvable in R.
b
x
a
Solvable Equations in R
x 9 is solvable in R.
The solutions are 3 and 3
2
x 3 is solvable in R.
2
The solutions are
3 and 3.
Solvable Equations in R
If p 0
ALGEBRAIC
EXPRESSIONS
Math Division, IMSP, UPLB
Learning Objectives
Upon completion, you should be able to
Give examples of algebraic
expressions; and
Perform operations involving integer
exponents.
2
Math Division, IMSP, UPLB
ALGEBRAIC
Introduction
Suppo
RATIONAL
EXPRESSIONS
Learning objectives
Upon completion, you should be able to
Simplify
rational expressions;
Perform
addition, subtraction,
multiplication and division on rational
expressions; and
Simplify
complex fractions.
2
RATIONAL EXPRESSIONS
Ra
OPERATIONS ON
POLYNOMIALS
OPERATIONS ON POLYNOMIALS
Learning Objectives
Upon completion you should be able to
perform the following operations on
polynomials:
ADDITION
SUBTRACTION
MULTIPLICATION
DIVISION
2
OPERATIONS ON POLYNOMIALS
POLYNOMIAL
is
an a
SPECIAL PRODUCTS AND
FACTORING
Special Products and Factoring
Upon completion, you should be able to
Find special products
Factor a polynomial completely
2
Special Products
rules for finding products in a faster way
long multiplication is the last resor
RATIONAL EXPONENTS &
RADICALS
OBJECTIVES
na
Upon completion, you should be able to
Define the principal nth root of numbers;
Simplify radicals; and
Perform addition, subtraction, multiplication
and division of radicals.
2
RATIONAL EXPONENTS & RADICALS
EXERCISES 1.2
1. Tell which of the axioms of the real numbers justifies each of
the following statements:
a.
b.
c.
d.
e.
2. True or false: If true, provide a proof. Otherwise, give a
counterexample.
a.
b. Subtraction is commutative.
c. Subtraction is asso
EXERCISES 1.1
20. The set of people who voted in the last election
Use both the roster and the rule method to specify the sets in
Problems 1 through 10.
1. The RGEP courses you are enrolled in this semester
2. The counting numbers less than 20
3. The frac
6/20/2012
NOTION OF A SET
Learning objectives:
Upon completion you should be able to:
Give examples of a well-defined set;
Name the elements of a set;
Describe a set using the roster and rule
methods;
Give examples of empty and universal
sets.
NOTION
Learning Objectives
At the end of the lesson, you should be able to
define subtraction and division operations
enumerate the properties of real numbers
illustrate the closure property for real numbers
identify the identity and inverse elements for
add
6/27/2012
Learning Objectives: upon completion,
you should be able to perform the
following operations:
Union
Intersection
Complement
Difference
Cross product
S
E
T
O
P
E
R
A
T
I
O
N
S
2
Uses a close region in the plane to
represent sets
U
A
B
C
S
E
T
O
Learning Objectives
Subsets of the Set of Real
Numbers
Mathematics Division, IMSP, UPLB
At the end of the lesson, you should be able to
identify subsets of the set of real numbers
recognize the various forms of rational
numbers
distinguish rational num
RATIONAL
EXPRESSIONS
Learning objectives
Upon completion, you should be able to
Simplify
rational expressions;
Perform
addition, subtraction,
multiplication and division on rational
expressions; and
Simplify
complex fractions.
2
RATIONAL EXPRESSIONS
Ra
SPECIAL PRODUCTS AND
FACTORING
Special Products and Factoring
Upon completion, you should be able to
Find special products
Factor a polynomial completely
2
Special Products
rules for finding products in a faster way
long multiplication is the last resor
OPERATIONS ON
POLYNOMIALS
OPERATIONS ON POLYNOMIALS
Learning Objectives
Upon completion you should be able to
perform the following operations on
polynomials:
ADDITION
SUBTRACTION
MULTIPLICATION
DIVISION
2
OPERATIONS ON POLYNOMIALS
POLYNOMIAL
is
an a
Equations in Linear and
Quadratic Forms
At the end of this section, you should be
able to solve equations that are convertible
to equations in linear or quadratic forms:
Equations involving rational expressions
Equations in quadratic forms
Equations in
ALGEBRAIC
EXPRESSIONS
MathDivision,IMSP,UPLB
LearningObjectives
Upon completion, you should be able to
Give examples of algebraic
expressions; and
Perform operations involving integer
exponents.
2
MathDivision,IMSP,UPLB
ALGEBRAIC
Introduction
Suppose we a
Word Problems
Objectives
Upon completion, you should be able to:
Translate English statements into
mathematical statements
Use the techniques learned in solving
linear, quadratic and systems of
equations in solving word problems
Check reasonableness of th
RATIONAL EXPONENTS &
RADICALS
OBJECTIVES
na
Upon completion, you should be able to
Define the principal nth root of numbers;
Simplify radicals; and
Perform addition, subtraction, multiplication
and division of radicals.
2
RATIONAL EXPONENTS & RADICALS
EQUATIONS &
INEQUALITIES
MATH 11
COLLEGE ALGEBRA
Chapter Outline
Equality Axioms
Linear and Quadratic Equations
Equations involving Radicals
Equations in Quadratic Form
Linear and Quadratic Inequalities
Word Problems
2
EQUATIONS & INEQUALITIES
Objectives
INEQUALITIES and
ABSOLUTE VALUE
MATH 11
College Algebra
Objectives
Upon completion, you should be able to
define order relations between numbers
enumerate order properties of
define
absolute value of a number and state its
properties
solve
linear, quadrat