Pre-Calculus Essentials

CHAPTER 1

Sets, Numbers, Operations, and Properties

1.1 Sets

A collection of objects categorized together is called a set. There are two standard ways to represent sets—the roster method and the set-builder method. For example,

{2, 3, 4}

is in roster form and

{x | x is a counting number between 1 and 5}

is in set-builder form, although both describe the same set.

The symbol ∈ is used to represent is an element of and the symbol ⊆ is used to represent is a subset of. Here are two important definitions concerning sets.

A ⊆ B if and only if every element of A is an element of B.

A = B if and only if A ⊆ B and B ⊆ A.

Two important set operations are union, denoted by ∪, and intersection, denoted by ∩, defined below.

A ∪ B = { x | x ∈ A or x ∈ B }

A ∩ B = { x | x ∈ A and x ∈ B }

1.2 Real Numbers and Their Components

Real numbers provide the basis for most pre-calculus mathematics topics. The set of all real numbers has various components. These components are the set of all natural numbers, N, the set of all whole numbers, W, the set of all integers, I, the set of all rational numbers, Q, and the set of all irrational numbers, S. Then,

and S = { x | x has a decimal name which is nonterminating and does not have a repeating block}.

It is obvious that N ⊆ W, W ⊆ I, and I ⊆ Q, but a similar relationship does not hold between Q and S. More specifically, the decimal names for elements of Q are

terminating or

nonterminating with a repeating block.

For example, e9780738670546_i0004.jpg and e9780738670546_i0005.jpg This means that Q and S have no common elements. Examples of irrational numbers include .101001000..., π, and √2.

All real numbers are normally represented by R, and R = Q ∪ S. This means that every real number is either rational or irrational. A nice way to visualize real numbers geometrically is to see that they can be put in a one-to-one correspondence with the set of all points on a line.

1.3 Real Number Properties of Equality

The standard properties of equality involving real numbers are:

Reflexive Property of Equality

For each real number a,

a = a.

Symmetric Property of Equality

For each real number a, for each real number b,

if a = b, then b = a.

Transitive Property of Equality

For each real number a, for each real number b, for each real number c,

if a = b and b = c, then a = c.

Other properties of equality are listed in Chapter 4.

1.4 Real Number Operations and Their Properties

The operations of addition and multiplication are of particular importance. As a result, many properties concerning those operations have been determined and named. Here is a list of the most important of these properties.

Closure Property of Addition

For every real number a, for every real number b,

a + b

is a real number.

Closure Property of Multiplication

For every real number a, for every real number b,

ab

is a real number.

Commutative Property of Addition

For every real number a, for every real number b,

a + b = b + a.

Commutative Property of Multiplication

For every real number a, for every real number b,

ab = ba.

Associative Property of Addition

For every real number a, for every real number b, for every real number c,

(a + b) + c = a + (b + c).

Associative Property of Multiplication

For every real number a, for every real number b, for every real number c,

(ab)c = a(bc).

Identity Property of Addition

For every real number a,

a + 0 = 0 + a = a.

Identity Property of Multiplication

For every real number a,

a × 1 = 1 × a = a.

Inverse Property of Addition

For every real number a, there is a real number − a such that

a + − a = − a + a = 0.

Inverse Property of Multiplication

For every real number a, a ≠ 0, there is a real number a−1 such that

a × a−1 = a−1 × a = 1.

Distributive Property

For every real number a, for every real number b, for every real number c,

a(b + c) = ab + ac.

The operations of subtraction and division are also important, but less important than addition and multiplication. Here are the definitions for these operations.

For every real number a, for every real number b, for every real number c,

a − b = c if and only if b + c = a.

For