Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321716811

9th Edition

5,201 Questions

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40,131 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the fundamental concepts of real numbers and set theory, emphasizing the importance of precise notation and the properties of numbers. Students learn about various sets of numbers, methods to represent them, and how to compute basic operations using established principles such as the order of operations, and properties like commutative, associative, and distributive laws. Additionally, the material connects these ideas with practical examples such as decimal approximations and fraction operations, highlighting both historical context and modern applications.

Learning Objectives

1

Describe and differentiate between various types of sets and number systems (natural, whole, integers, rational, irrational, and real numbers).

2

Explain and apply set notation, including the roster and set-builder methods, as well as operations such as union, intersection, and complement.

3

Demonstrate how to evaluate numerical expressions using the order of operations and the properties of real numbers.

4

Compute least common multiples and use them to add and subtract fractions.

5

Understand and apply the properties of equality (reflexive, symmetric, transitive, substitution) and arithmetic properties (commutative, associative, distributive, identity, inverse) in problem-solving.

Key Concepts

CONCEPT

DEFINITION

Set

A well-defined collection of distinct objects. The objects in a set are called its elements.

Empty Set

A set with no elements, denoted by {} or the symbol ∅.

Roster Method

A way to denote a set by listing its elements within braces.

Set-Builder Notation

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

Subset

A set A is a subset of B if every element of A is also in B, denoted by A ⊆ B.

Intersection

The set of elements that belong to both sets A and B, denoted by A ∩ B.

Union

The set of elements that are in A or B (or both), denoted by A ∪ B.

Complement

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

Rational Number

A number that can be expressed as a quotient a/b, where a and b are integers and b ≠ 0. Its decimal representation terminates or repeats.

Irrational Number

A number that cannot be written as a quotient of integers; its decimal representation is non-terminating and non-repeating.

Real Numbers

The union of the set of rational numbers with the set of irrational numbers.

Order of Operations

The rules (PEMDAS/BODMAS) that determine the sequence in which operations are performed in an expression.

Properties of Real Numbers

Include commutative, associative, distributive, identity, and inverse properties used to simplify expressions and solve equations.

Least Common Multiple (LCM)

The smallest positive integer that is a multiple of each of two or more numbers.

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

True or False Rational numbers have decimals that either terminate or are nonterminating with a repeating block of digits.
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Step-by-Step Explanations

QUESTION

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

STEP-BY-STEP ANSWER:

Step 1: Write down the elements in set A and set B.
Step 2: Identify the common elements in both sets (intersection). For A and B, the common elements are 3, 5, and 8.
Step 3: Write A ∩ B = {3, 5, 8}.
Step 4: For the union, combine all distinct elements from both sets: 1, 3, 5, 7, 8.
Step 5: Write A ∪ B = {1, 3, 5, 7, 8}.
Final Answer: A ∩ B = {3, 5, 8} and A ∪ B = {1, 3, 5, 7, 8}.

Finding the Union and Intersection of Sets

QUESTION

Approximate 20.98752 to two decimal places using (a) truncation and (b) rounding.

STEP-BY-STEP ANSWER:

Step 1: Identify the digit at the second decimal place; here it is 8 in 20.98.
Step 2: For truncation, simply drop all digits after 8. So the truncated value is 20.98.
Step 3: For rounding, look at the digit immediately after the second decimal place. In 20.98752, this digit is 7.
Step 4: Since 7 is 5 or more, add 1 to the digit in the second decimal place: 8 becomes 9.
Step 5: The rounded value is 20.99.
Final Answer: Truncated = 20.98; Rounded = 20.99.

Approximating Decimals by Truncating and Rounding

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Common Mistakes

  • Repeating elements in a set when using the roster method, forgetting that order does not matter.
  • Confusing the union and intersection of sets, mistakenly combining operations.
  • Misapplying the order of operations by performing addition before multiplication.
  • Incorrectly rounding decimals by not checking the digit immediately after the specified decimal place.
  • Overlooking the importance of common denominators when adding or subtracting fractions.