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
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