Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321979476

10th Edition

5,609 Questions

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320,854 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the fundamental idea of sets as collections of distinct objects and explores various methods to represent them, including the roster method and set-builder notation. Key operations such as union, intersection, and complement are defined and illustrated. In addition, the section reviews the classification of numbers—natural, whole, integers, rational, and irrational—and covers essential properties and operations on real numbers, ensuring students understand both the theoretical and practical aspects of working with mathematical sets and expressions.

Learning Objectives

1

Identify and work with sets using both the roster method and set-builder notation.

2

Compute and interpret set operations such as union, intersection, and complement.

3

Classify various types of numbers (natural, whole, integers, rational, and irrational) and understand their relationships.

4

Apply the order of operations and properties of real numbers to evaluate numerical expressions and simplify algebraic statements.

5

Utilize mathematical representations (such as Venn diagrams) to visualize relationships among sets.

Key Concepts

CONCEPT

DEFINITION

Set

A well-defined collection of distinct objects, where each object is called an element.

Element

An object or member of a set.

Empty Set (Null Set)

A set that contains no elements, denoted by the symbol ∅.

Roster Method

A method of denoting a set by listing its elements within braces.

Set-builder Notation

A method of describing a set by stating the properties that its members satisfy.

Union (A ∪ B)

The set consisting of all elements that are in A, or in B, or in both.

Intersection (A ∩ B)

The set of all elements that are common to both sets A and B.

Complement

For a set A, the complement consists of all elements in the universal set that are not in A.

Subset (A ⊆ B)

A set A is a subset of B if every element of A is also an element of B.

Order of Operations

A set of rules (PEMDAS/BODMAS) that dictate the sequence in which operations should be performed in a numerical expression.

Example Problems

Example 1

The numbers in the set $\left\{x | x=\frac{a}{b}\right.$, where $a, b$ are integers and $b \neq 0\}$ are called ______ numbers

Example 2

The value of the expression $4+5 \cdot 6-3$ is ____

Example 3

The fact that $2 x+3 x=(2+3) x$ is a consequence of the ______ Property

Example 4

"The product of 5 and $x+3$ equals 6 " may be written as ________.

Example 5

The intersection of sets $A$ and $B$ is denoted by which of the following? $$ \begin{array}{llll}{\text { (a) } A \cap B} & {\text { (b) } A \cup B} & {\text { (c) } A \subseteq B} & {\text { (d) } A \varnothing B}\end{array} $$

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Step-by-Step Explanations

QUESTION

Given A = {1, 3, 5, 8} and B = {3, 5, 7, 6}, find A ∩ B.

STEP-BY-STEP ANSWER:

Step 1: List the elements in set A: {1, 3, 5, 8}.
Step 2: List the elements in set B: {3, 5, 7, 6}.
Step 3: Identify the common elements present in both sets.
Step 4: The common elements are 3 and 5.
Final Answer: A ∩ B = {3, 5}.

Intersection of Sets

QUESTION

Evaluate the expression 2 + 3 × 6 following the correct order of operations.

STEP-BY-STEP ANSWER:

Step 1: Identify the multiplication and addition operations. Since multiplication must be done first, compute 3 × 6.
Step 2: 3 × 6 = 18.
Step 3: Now add 2 to the product: 2 + 18.
Final Answer: 2 + 3 × 6 = 20.

Evaluating a Numerical Expression

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Common Mistakes

  • Listing repeated elements in a set, despite the requirement for distinct objects.
  • Confusing the union (elements in either set) with the intersection (elements common to both sets).
  • Forgetting to apply the proper order of operations, often performing addition before multiplication.
  • Misusing set-builder notation by not specifying a clear property that defines the set.
  • Neglecting the use of parentheses in complex expressions, which can lead to calculation errors.