Download Honors Pre-Calculus

Honors Pre-Calculus
Appendix A1
Algebra Essentials
Objectives
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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:
𝐷 = 0,1,2,3,4,5,6,7,8,9
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:
𝐷 = 𝑥 𝑥 is a digit
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
– 𝐸 = 𝑥|𝑥 is an even digit = 0,2,4,6,8
• (b) The set of odd digits
– O= 𝑥|𝑥 is an odd digit = 1,3,5,7,9
Intersection and Union
• If A and B are sets.
• The intersection of A with B, denoted by 𝐴 ∩
𝐵 is the set consisting of elements that belong
to both A and B.
• The union of A with B, denoted by 𝐴 ∪ 𝐵, is
the set consisting of elements that belong to
either A or B, or both.
Finding the Intersection and Union of
Sets
Let 𝐴 = 1,2,4,6 , 𝐵 = {2,4,7}, and 𝐶 = 3,5,6 .
Find:
(a) 𝐴 ∩ 𝐵
𝐴 ∩ 𝐵 = 1,2,4,6 ∩ 2,4,7 = 2,4
(b) 𝐴 ∪ 𝐵
𝐴 ∪ 𝐵 = 1,2,4,6 ∪ 2,4,7 = 1,2,4,6,7
(c) 𝐵 ∩ 𝐴 ∪ 𝐶
𝐵 ∩ 𝐴 ∪ 𝐶 = 2,4,7 ∩ 1,2,4,6 ∪ 3,5,6
= 2,4,7 ∩ 1,2,3,4,5,6 = {2,4}
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
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
x
x 2  16
• is {x | x  4}
since if x=4 or x=-4 the denominator is not 0,
so this expression is defined for all numbers.
Domain (continued)
• Example 2
The domain of the variable x in the expression
x
x 2  16
• is
since if x=4 or x=-4 the denominator is not 0,
so this expression is defined for all numbers.
Homework
• Pg A11 9-14, 67-78