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Contemporary Mathematics

1.3 Understanding Venn Diagrams

Contemporary Mathematics1.3 Understanding Venn Diagrams

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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
In a residence, a cat is looking at tools, an instruction booklet for assembly, and pieces of furniture laid out on the floor.
Figure 1.6 When assembling furniture, instructions with images are easier to follow, just like how set relationships are easier to understand when depicted graphically. (credit: "Time to assemble more Ikea furniture!" by Rod Herrea/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

  1. Utilize a universal set with two sets to interpret a Venn diagram.
  2. Utilize a universal set with two sets to create a Venn diagram.
  3. Determine the complement of a set.

Have you ever ordered a new dresser or bookcase that required assembly? When your package arrives you excitedly open it and spread out the pieces. Then you check the assembly guide and verify that you have all the parts required to assemble your new dresser. Now, the work begins. Luckily for you, the assembly guide includes step-by-step instructions with images that show you how to put together your product. If you are really lucky, the manufacturer may even provide a URL or QR code connecting you to an online video that demonstrates the complete assembly process. We can likely all agree that assembly instructions are much easier to follow when they include images or videos, rather than just written directions. The same goes for the relationships between sets.

Interpreting Venn Diagrams

Venn diagrams are the graphical tools or pictures that we use to visualize and understand relationships between sets. Venn diagrams are named after the mathematician John Venn, who first popularized their use in the 1880s. When we use a Venn diagram to visualize the relationships between sets, the entire set of data under consideration is drawn as a rectangle, and subsets of this set are drawn as circles completely contained within the rectangle. The entire set of data under consideration is known as the universal set.

Consider the statement: All trees are plants. This statement expresses the relationship between the set of all plants and the set of all trees. Because every tree is a plant, the set of trees is a subset of the set of plants. To represent this relationship using a Venn diagram, the set of plants will be our universal set and the set of trees will be the subset. Recall that this relationship is expressed symbolically as: TreesPlants.TreesPlants. To create a Venn diagram, first we draw a rectangle and label the universal set “U=Plants.U=Plants.” Then we draw a circle within the universal set and label it with the word “Trees.”

A single-set Venn diagram is shaded. Outside the set, it is labeled as 'Trees.' Outside the Venn diagram, 'U equals Plants' is labeled.
Figure 1.7

This section will introduce how to interpret and construct Venn diagrams. In future sections, as we expand our knowledge of relationships between sets, we will also develop our knowledge and use of Venn diagrams to explore how multiple sets can be combined to form new sets.

Example 1.18

Interpreting the Relationship between Sets in a Venn Diagram

Write the relationship between the sets in the following Venn diagram, in words and symbolically.

A single-set Venn diagram is shaded. Outside the set, it is labeled as 'T equals Terriers.' Outside the Venn diagram, 'Parallelograms' is labeled.
Figure 1.8

Your Turn 1.18

1.
Write the relationship between the sets in the following Venn diagram, in words and symbolically.
A single-set Venn diagram is shaded. Outside the set, it is labeled as 'L equals Lions.' Outside the Venn diagram, 'Parallelograms' is labeled.
Figure 1.9

So far, the only relationship we have been considering between two sets is the subset relationship, but sets can be related in other ways. Lions and tigers are both different types of cats, but no lions are tigers, and no tigers are lions. Because the set of all lions and the set of all tigers do not have any members in common, we call these two sets disjoint sets, or non-overlapping sets.

Two sets AA and BB are disjoint sets if they do not share any elements in common. That is, if aa is a member of set AA, then aa is not a member of set BB. If bb is a member of set BB, then bb is not a member of set AA. To represent the relationship between the set of all cats and the sets of lions and tigers using a Venn diagram, we draw the universal set of cats as a rectangle and then draw a circle for the set of lions and a separate circle for the set of tigers within the rectangle, ensuring that the two circles representing the set of lions and the set of tigers do not touch or overlap in any way.

A two-set Venn diagram not intersecting one another is given. Outside the Venn diagram, 'U equals Cats' is labeled. The first set is labeled T equals tigers while the second set is labeled L equals lions.
Figure 1.10

Example 1.19

Describing the Relationship between Sets

Describe the relationship between the sets in the following Venn diagram.

A two-set Venn diagram not intersecting one another is given. Outside the Venn diagram, 'U equals 2D Figures' is labeled. The first set is labeled T equals Triangles while the second set is labeled S equals Squares.
Figure 1.11

Your Turn 1.19

1.

Describe the relationship between the sets in the following Venn diagram.

A two-set Venn diagram not intersecting one another is given. Outside the Venn diagram, 'U equals Birds' is labeled. The first set is labeled E equals Eagles while the second set is labeled C equals the Canaries.
Figure 1.12

Creating Venn Diagrams

The main purpose of a Venn diagram is to help you visualize the relationship between sets. As such, it is necessary to be able to draw Venn diagrams from a written or symbolic description of the relationship between sets.

Procedure

To create a Venn diagram:

  1. Draw a rectangle to represent the universal set, and label it U=set nameU=set name.
  2. Draw a circle within the rectangle to represent a subset of the universal set and label it with the set name.

Checkpoint

If there are multiple disjoint subsets of the universal set, their separate circles should not touch or overlap.

Example 1.20

Drawing a Venn Diagram to Represent the Relationship Between Two Sets

Draw a Venn diagram to represent the relationship between each of the sets.

  1. All rectangles are parallelograms.
  2. All women are people.

Your Turn 1.20

1.
Draw a Venn diagram to represent the relationship between each of the sets. All natural numbers are integers.
2.
A \subset U. Draw a Venn diagram to represent this relationship.

Example 1.21

Drawing a Venn Diagram to Represent the Relationship Between Three Sets

All bicycles and all cars have wheels, but no bicycle is a car. Draw a Venn diagram to represent this relationship.

Your Turn 1.21

1.
Airplanes and birds can fly, but no birds are airplanes. Draw a Venn diagram to represent this relationship.

The Complement of a Set

Recall that if set AA is a proper subset of set UU, the universal set (written symbolically as AUAU), then there is at least one element in set UU that is not in set AA. The set of all the elements in the universal set UU that are not in the subset AA is called the complement of set AA, A'A'. In set builder notation this is written symbolically as: A'={xU|xA}.A'={xU|xA}. The symbol is used to represent the phrase, “is a member of,” and the symbol is used to represent the phrase, “is not a member of.” In the Venn diagram below, the complement of set AA is the region that lies outside the circle and inside the rectangle. The universal set UU includes all of the elements in set AA and all of the elements in the complement of set AA, and nothing else.

A single-set Venn diagram is shaded. Outside the set, it is labeled as 'A.' Outside the Venn diagram, 'U' is labeled.
Figure 1.18

Consider the set of digit numbers. Let this be our universal set, U={0,1,2,3,4,5,6,7,8,9}.U={0,1,2,3,4,5,6,7,8,9}. Now, let set AA be the subset of UU consisting of all the prime numbers in set UU, A={2,3,5,7}.A={2,3,5,7}. The complement of set AA is A'={0,1,4,6,8,9}.A'={0,1,4,6,8,9}. The following Venn diagram represents this relationship graphically.

A single-set Venn diagram is labeled 'A equals (2, 3, 5, 7).'
Figure 1.19

Example 1.22

Finding the Complement of a Set

For both of the questions below, AA is a proper subset of UU.

  1. Given the universal set U={Billie Eilish, Donald Glover, Bruno Mars, Adele, Ed Sheeran}U={Billie Eilish, Donald Glover, Bruno Mars, Adele, Ed Sheeran} and set A={Donald Glover, Bruno Mars, Ed Sheeran}A={Donald Glover, Bruno Mars, Ed Sheeran}, find A'.A'.
  2. Given the universal set U={d|d is a dog}U={d|d is a dog} and B={b ∈ U|b is a beagle}B={b ∈ U|b is a beagle}, find B'.B'.

Your Turn 1.22

For both of the questions below, A is a proper subset of U.
1.
Given the universal set U = \left\{ {{\text{red, orange, yellow, green, blue, indigo, violet}}} \right\} and set A = \left\{ {{\text{yellow, red, blue}}} \right\}, find A'.
2.
Given the universal set U = \{ c|c\,{\text{is a cat\} }} and set A = \{ c \in U|c{\text{ is not a lion\}, }} find A'.

Check Your Understanding

15.
A Venn diagram is a graphical representation of the _____________ between sets.
16.
In a Venn diagram, the set of all data under consideration, the _____________ set, is drawn as a rectangle.
17.
Two sets that do not share any elements in common are _____________ sets.
18.
The _____________ of a subset A or the universal set, U, is the set of all members of U that are not in A.
19.
The sets A and A' are _____________ subsets of the universal set.

Section 1.3 Exercises

For the following exercises, interpret each Venn diagram and describe the relationship between the sets, symbolically and in words.
1.
A single-set Venn diagram is shaded. Outside the set, it is labeled as 'T equals Team sports.' Outside the Venn diagram, 'U equals Sports' is labeled.
2.
A single-set Venn diagram is shaded. Outside the set, it is labeled as 'A equals Apples.' Outside the Venn diagram, 'U equals Fruit' is labeled.
3.
A single-set Venn diagram is shaded. Outside the set is labeled 'P equals Pencils.' Outside the Venn diagram, 'U equals Writing Utensils' is labeled.
4.
A single-set Venn diagram is labeled 'B equals Board Games'. Outside the Venn diagram, it is labeled as 'U equals Games.'
5.
A two-set Venn diagram not intersecting one another is given. Outside the Venn diagram, 'U equals Writing Utensils' is labeled.
6.
A two-set Venn diagram not intersecting one another is given. Outside the Venn diagram, 'U equals Fruit' is labeled.
7.
A two-set Venn diagram not intersecting one another is given. Outside the Venn diagram, it is labeled as 'U equals Writing Games.' The first set is labeled A equals Card games while the second set is labeled B equals Video Games.
8.
A two-set Venn diagram not intersecting one another is given. Outside the Venn diagram, it is labeled as 'U equals Investments.' The first set is labeled A equals Stocks while the second set is labeled B equals Bonds.
For the following exercises, create a Venn diagram to represent the relationships between the sets.
9.
All birds have wings.
10.
All cats are animals.
11.
All almonds are nuts, and all pecans are nuts, but no almonds are pecans.
12.
All rectangles are quadrilaterals, and all trapezoids are quadrilaterals, but no rectangles are trapezoids.
13.
Lizards \subset Reptiles.
14.
Ladybugs \subset Insects.
15.
Ladybugs \subset Insects and Ants \subset Insects, but no Ants are Ladybugs.
16.
Lizards \subset Reptiles and Snakes \subset Reptiles, but no Lizards are Snakes.
17.
A and B are disjoint subsets of U.
18.
C and D are disjoint subsets of U.
19.
T is a subset of U.
20.
S is a subset of U.
21.
J = Jazz, M = Music, and J \subset M.
22.
R Reggae, M Music, and R \subset M.
23.
J = Jazz, R = Reggae, and M = Music are sets with the following relationships: J \subset M, R \subset M, and R is disjoint from J.
24.
J = Jazz, B = Bebop, and M = Music are sets with the following relationships: J \subset M and B \subset J.
For the following exercises, the universal set is the set of single digit numbers, U = \{ 0,1,2,3,4,5,6,7,8,9\} . Find the complement of each subset of U.
25.
A = \{ 6,7,8\}
26.
A = \{ 0,2,4,6,8\}
27.
A = \{ {\text{ }}\}
28.
A = \{ 0,1,4,6,8,9\}
29.
A = \{ 0,1,2,3,4,5,6,7,8,9\}
30.
A = \{ 0,1,3,4,5,6,7,9\}
31.
A = \{ 1,2,3,4,5,6,7,8,9\}
32.
A = \{ 0,3,6,9\}
For the following exercises, the universal set is U = {Bashful, Doc, Dopey, Grumpy, Happy, Sleepy, Sneezy}. Find the complement of each subset of U.
33.
A = {Happy, Bashful, Grumpy}
34.
A = {Sleepy, Sneezy}
35.
A = {Doc}
36.
A = {Doc, Dopey}
37.
A = \emptyset
38.
A = {Doc, Grumpy, Happy, Sleepy, Bashful, Sneezy, Dopey}
For the following exercises, the universal set is U = \mathbb{N} = \{ 1,2,3, \ldots \} . Find the complement of each subset of U.
39.
A = \{ 1,2,3,4,5\}
40.
A = \{ 1,3,5, \ldots \}
41.
A = \{ 1\}
42.
A = \{ 4,5,6, \ldots \}
For the following exercises, use the Venn diagram to determine the members of the complement of set A,{\text{ }}A'.
43.
A single-set Venn diagram is labeled 'A equals (2, 3, 4, 9).' Outside the Venn diagram, the union of the Venn diagram is marked 'U equals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).'
44.
A single-set Venn diagram is labeled 'A equals (2, 3, 4, 9).' Outside the Venn diagram, the union of the Venn diagram is marked 'U equals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).'
45.
A single-set Venn diagram is labeled A equals (n, e, t). Outside the Venn diagram, the union of the Venn diagram is marked 'U equals (l, i, s, t, e, n).'
46.
A single-set Venn diagram is labeled 'A equals (l, i, n, e, s).' Outside the Venn diagram, the union of the Venn diagram is marked 'U equals (l, i, s, t, e, n).'
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