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Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index
Soccer players are shown on a field and one of them is about to kick the ball.
Figure 1.4 The players on a soccer team who are actively participating in a game are a subset of the greater set of team members. (Credit: “PAFC-Mezokovesd-108” by Puskás Akadémia/Flickr, Public Domain Mark 1.0)

Learning Objectives

After completing this section, you should be able to:

  1. Represent subsets and proper subsets symbolically.
  2. Compute the number of subsets of a set.
  3. Apply concepts of subsets and equivalent sets to finite and infinite sets.

The rules of Major League Soccer (MLS) allow each team to have up to 30 players on their team. However, only 18 of these players can be listed on the game day roster, and of the 18 listed, 11 players must be selected to start the game. How the coaches and general managers form the team and choose the starters for each game will determine the success of the team in any given year.

The entire group of 30 players is each team’s set. The group of game day players is a subset of the team set, and the group of 11 starters is a subset of both the team set and the set of players on the game day roster.

Set AA is a subset of set BB if every member of set AA is also a member of set BB. Symbolically, this relationship is written as ABAB.

Sets can be related to each other in several different ways: they may not share any members in common, they may share some members in common, or they may share all members in common. In this section, we will explore the way we can select a group of members from the whole set.

Checkpoint

Every set is also a subset of itself, BBBB

Recall the set of flatware in our kitchen drawer from Section 1.1, F={fork, spoon, knife, meat thermometer, can opener}F={fork, spoon, knife, meat thermometer, can opener}. Suppose you are preparing to eat dinner, so you pull a fork and a knife from the drawer to set the table. The set D={knife, fork}D={knife, fork} is a subset of set FF, because every member or element of set DD is also a member of set FF. More specifically, set DD is a proper subset of set FF, because there are other members of set FF not in set DD. This is written as DFDF. The only subset of a set that is not a proper subset of the set would be the set itself.

Checkpoint

The empty set or null set, , is a proper subset of every set, except itself.

Graphically, sets are often represented as circles. In the following graphic, set AA is represented as a circle completely enclosed inside the circle representing set BB, showing that set AA is a proper subset of set BB. The element xx represents an element that is in both set AA and set BB.

A two-step Venn diagram, A and B, is shown, where A is inside B. A is marked at its center with an 'x.'
Figure 1.5

Checkpoint

While we can list all the subsets of a finite set, it is not possible to list all the possible subsets of an infinite set, as it would take an infinitely long time.

Example 1.11

Listing All the Proper Subsets of a Finite Set

Set LL is a set of reading materials available in a shop at the airport, L={newspaper, magazine, book}L={newspaper, magazine, book}. List all the subsets of set LL.

Your Turn 1.11

1.
Consider the set of possible outcomes when you flip a coin, S = { heads, tails } . List all the possible subsets of set S .

Example 1.12

Determining Whether a Set Is a Proper Subset

Consider the set of common political parties in the United States, P={Democratic, Green, Libertarian, Republican}P={Democratic, Green, Libertarian, Republican}. Determine if the following sets are proper subsets of PP.

  1. M={Democratic, Republican}M={Democratic, Republican}
  2. G={Green}G={Green}
  3. V={Republican, Libertarian, Green, Democratic}V={Republican, Libertarian, Green, Democratic}

Your Turn 1.12

Consider the set of generation I legendary Pokémon, L = { Articuno, Zapdos, Moltres, Mewtwo } . Give an example of a proper subset containing:

1.
one member.
2.
three members.
3.
no members.

Example 1.13

Expressing the Relationship between Sets Symbolically

Consider the subsets of a standard deck of cards: S={spades, hearts, diamonds, clubs}S={spades, hearts, diamonds, clubs}; R={hearts, diamonds}R={hearts, diamonds}; B={spades, clubs}B={spades, clubs}; and C={clubs}C={clubs}.

Express the relationship between the following sets symbolically.

  1. Set SS and set BB.
  2. Set CC and set BB.
  3. Set RR and RR.

Your Turn 1.13

1.
Express the relationship between the set of natural numbers, N = { 1 , 2 , 3 , } , and the set of even numbers, E = { 2 , 4 , 6 , } .

Exponential Notation

So far, we have figured out how many subsets exist in a finite set by listing them. Recall that in Example 1.11, when we listed all the subsets of the three-element set L={newspaper, magazine, book}L={newspaper, magazine, book} we saw that there are eight subsets. In Your Turn 1.11, we discovered that there are four subsets of the two-element subset, S={heads, tails}S={heads, tails}. A one-element set has two subsets, the empty set and itself. The only subset of the empty set is the empty set itself. But how can we easily figure out the number of subsets in a very large finite set? It turns out that the number of subsets can be found by raising 2 to the number of elements in the set, using exponential notation to represent repeated multiplication. For example, the number of subsets of the set L={newspaper, magazine, book}L={newspaper, magazine, book} is equal to 23=222=823=222=8. Exponential notation is used to represent repeated multiplication, bn=bbbbbn=bbbb, where bb appears as a factor nn times.

FORMULA

The number of subsets of a finite set AA is equal to 2 raised to the power of n(A)n(A), where n(A)n(A) is the number of elements in set AA: Number of Subsets of Set A=2n(A)Number of Subsets of Set A=2n(A).

Checkpoint

Note that 20=120=1, so this formula works for the empty set, also.

Example 1.14

Computing the Number of Subsets of a Set

Find the number of subsets of each of the following sets.

  1. The set of top five scorers of all time in the NBA: S={ LeBron James, Kareem Abdul-Jabbar, Karl Malone, Kobe Bryant, Michael Jordan }.S={ LeBron James, Kareem Abdul-Jabbar, Karl Malone, Kobe Bryant, Michael Jordan }.
  2. The set of the top four bestselling albums of all time: A={ Thriller, Hotel California, The Beatles White Album, Led Zepplin IV }A={ Thriller, Hotel California, The Beatles White Album, Led Zepplin IV }.
  3. R={ Snap, Crackle, Pop }R={ Snap, Crackle, Pop }.

Your Turn 1.14

1.
Compute the total number of subsets in the set of the top nine tennis grand slam singles winners, T = { Margaret Court, Serena Williams, Steffi Graff, Rafael Nadal, Novak Djokovic,  Roger Federer, Helen Wills, Martina Navratilova, Chris Everett } .

Equivalent Subsets

In the early 17th century, the famous astronomer Galileo Galilei found that the set of natural numbers and the subset of the natural numbers consisting of the set of square numbers, n2n2, are equivalent. Upon making this discovery, he conjectured that the concepts of less than, greater than, and equal to did not apply to infinite sets.

Sequences and series are defined as infinite subsets of the set of natural numbers by forming a relationship between the sequence or series in terms of a natural number, nn. For example, the set of even numbers can be defined using set builder notation as {a|a=2nwherenis a natural number}{a|a=2nwherenis a natural number}. The formula in this case replaces every natural number with two times the number, resulting in the set of even numbers, {2,4,6,}{2,4,6,}. The set of even numbers is also equivalent to the set of natural numbers.

Who Knew?

Employment Opportunities

You can make a career out of working with sets. Applications of equivalent sets include relational database design and analysis.

Relational databases that store data are tables of related information. Each row of a table has the same number of columns as every other row in the table; in this way, relational databases are examples of set equivalences for finite sets. In a relational database, a primary key is set up to identify all related information. There is a one-to-one relationship between the primary key and any other information associated with it.

Database design and analysis is a high demand career with a median entry-level salary of about $85,000 per year, according to salary.com.

Example 1.15

Writing Equivalent Subsets of an Infinite Set

Using natural numbers, multiples of 3 are given by the sequence {3, 6, 9, …}{3, 6, 9, …}. Write this set using set builder notation by expressing each multiple of 3 using a formula in terms of a natural number, nn.

Your Turn 1.15

1.
Using natural numbers, multiples of 5 are given by the sequence { 5, 10, 15,  } . Write this set using set builder notation by associating each multiple of 5 in terms of a natural number, n .

Example 1.16

Creating Equivalent Subsets of a Finite Set That Are Not Equal

A fast-food restaurant offers a deal where you can select two options from the following set of four menu items for $6: a chicken sandwich, a fish sandwich, a cheeseburger, or 10 chicken nuggets. Javier and his friend Michael are each purchasing lunch using this deal. Create two equivalent, but not equal, subsets that Javier and Michael could choose to have for lunch.

Your Turn 1.16

1.
Serena and Venus Williams walk into the same restaurant as Javier and Michael, but they order the same pair of items, resulting in equal sets of choices. If Venus ordered a fish sandwich and chicken nuggets, what did Serena order?

Example 1.17

Creating Equivalent Subsets of a Finite Set

A high school volleyball team at a small school consists of the following players: {Angie, Brenda, Colleen, Estella, Maya, Maria, Penny, Shantelle}. Create two possible equivalent starting line-ups of six players that the coach could select for the next game.

Your Turn 1.17

1.
Consider the same group of volleyball players from above: {Angie, Brenda, Colleen, Estella, Maya, Maria, Penny, Shantelle}. The team needs to select a captain and an assistant captain from their members. List two possible equivalent subsets that they could select.

Check Your Understanding

8.
Every member of a __________ of a set is also a member of the set.
9.
Explain what distinguishes a proper subset of a set from a subset of a set.
10.
The __________ set is a proper subset of every set except itself.
11.
Is the following statement true or false? A A .
12.
If the cardinality of set A is n ( A ) = 10 , then set A has a total of ___________ subsets.
13.
Set A is ______________ to set B if n ( A ) = n ( B ) .
14.

If every member of set A is a member of set B and every member of set B is also a member set A , then set A is ____________ to set B .

For the following exercises, list all the proper subsets of each set.
1 .
{ chocolate, vanilla, strawberry }
2 .
{ true, false }
3 .
{ mother, father, daughter, son }
4 .
{ 7 }

For the following exercises, determine the relationship between the two sets and write the relationship symbolically.

D = { 0 , 1 , 2 , , 9 } , A = { 0 , 2 , 4 , 6 , 8 } , B = { 1 , 3 , 5 , 7 , 9 } , C = { 8 , 6 , 4 , 2 , 0 } , Z = { 0 } , and

5 .
D and A
6 .
B and D
7 .
C and D
8 .
Z and C
9 .
Z and
10 .
A and B
11 .
A and C
12 .
and D
13 .
B and C
14 .
A and Z
For the following exercises, calculate the total number of subsets of each set.
15 .
{ Adele, Beyonce, Cher, Madonna, Shakira }
16 .
{ Art, Paul }
17 .
{ Peter, Paul, Mary }
18 .
19 .
{ 3 }
20 .
{ l , o , v , e }
21 .
{   }
22 .
{ football, baseball, basketball, soccer, hockey, tennis, golf }
23 .
Set A , if n ( A ) = 12.
24 .
Set B , if n ( B ) = 9.
For the following exercises, use the set of letters in the word largest as the set, U = { l, a, r, g, e, s, t } .
25 .
Find a subset of U that is equivalent, but not equal, to the set: { l, a, s, t } .
26 .
Find a subset of U that is equal to the set: { l, a, s, t } .
27 .
Find a subset of U that is equal to the set: { a, r, t } .
28 .
Find a subset of U that is equivalent, but not equal, to the set { a, r, t, s } .
29 .
Find a subset of U that is equivalent, but not equal, to the set: { r, a, t, e, s } .
30 .
Find a subset of U that is equal to the set: { r, a, t, e, s } .
31 .
Find two three-character subsets of set U that are equivalent, but not equal, to each other.
32 .
Find two three-character subsets of set U that are equal to each other.
33 .
Find two five-character subsets of set U that are equal to each other.
34 .
Find two five-character subsets of set U that are equivalent, but not equal, to each other.
For the following exercises, use the set of integers as the set U = Z = { , 2 , 1 , 0 , 1 , 2 , } .
35 .
Find two equivalent subset of U with a cardinality of 7.
36 .
Find two equal subsets of U with a cardinality of 4.
37 .
Find a subset of U that is equivalent, but not equal to, { 0 , 3 , 6 , 9 , } .
38 .
Find a subset of U that is equivalent, but not equal to, { 1 , 4 , 9 , 16 , 25 , } .
39 .
True or False. The set of natural numbers, N = { 1 , 2 , 3 , } , is equivalent to set U .
40 .
True or False. Set U is an equivalent subset of the set of rational numbers, Q = { p q | p  and  q  are integers and  q 0. } .
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