Honors Pre-Calculus

Appendix A1Algebra Essentials

Objectives

• Work with sets• Graph Inequalities• Find Distance on the Real Number Line• Evaluate Algebraic Expressions• Determine the Domain of a Variable• Use the Laws of Exponents• Evaluate Square Roots• Use a Calculator to Evaluate Exponents

Working with Sets

• A set is a well-defined collection of distinct objects. The objects of a set are called its elements. By well-defined, we mean that there is a rule that enables us to determine whether a given object is an element of the set. If a set has no elements, it is called the empty set, or null set, and is denoted by

Examples of sets

• The set of digits:

In this notation, the braces {…} are used to enclose the elements in the set. This method of denoting sets is called roster notation.

• A second way to denote a set is to use set-builder notation, where this set would be written as:

This would be read as “D is the set of all x such that x is a digit.

Using Set-builder Notation

• Use Set-Builder and Roster Notation to denote the following sets.

• (a) The set of even digits

• (b) The set of odd digits– O

Intersection and Union

• If A and B are sets. • The intersection of A with B, denoted by is

the set consisting of elements that belong to both A and B.

• The union of A with B, denoted by , is the set consisting of elements that belong to either A or B, or both.

Finding the Intersection and Union of Sets

Let , , and . Find:(a)

(b)

(c)

Sets of Numbers

Complex Numbers

Symbols for Number Sets: Natural Numbers (Counting Numbers)

: Integers (from zahlen German for numbers)

: Rational Numbers (from quotient)

: Real Numbers

: Complex Numbers

Nine Zulu Queens Rule China

Closure• A numerical set is said to be closed under a

given operation if when that operation is performed on any element in the set the result of that operation is in that set.

• For example {x|x is even} is closed under addition because an even number plus an even number is even.

• {x|x is odd} is not closed under addtion because an odd number plus an odd odd number is not an odd number.

Closure

• Natural Numbers are closed under addition• Integers are closed under addition and

subtraction• Rational and Real Numbers are closed under

addition, subtraction, multiplication, and division (except 0).

• Complex numbers closed under addition, subtraction, multiplication, division (except 0), and taking roots.

Domain

• The set of values that a variable may assume is called the domain of the variable.The domain of the variable x in the expression

• is since if x=4 or x=-4 the denominator is not 0, so this expression is defined for all numbers.

2 16

x

x { | 4}x x

Domain (continued)

• Example 2The domain of the variable x in the expression

• is since if x=4 or x=-4 the denominator is not 0, so this expression is defined for all numbers.

2 16

x

x

Homework

• Pg A11 9-14, 67-78