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Contemporary Mathematics

Chapter 1

Contemporary MathematicsChapter 1
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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

Your Turn

1.1
1.
One possible solution: T = \{ {\text{wrench, screwdriver, hammer, plyers}}\} .
1.2
1.
This is not a well-defined set.
2.
This is a well-defined set.
1.3
1.
\emptyset {\text{ or \{ \} }}
1.4
1.
D = \{ 0,1,2, \ldots ,9\}
1.5
1.
M = \{ 1,3,5, \ldots \}
1.6
1.
C = \{c|c{\text{ }}{\text{is a car}}\}
1.7
1.
I = \{i|i{\text{ }}{\text{is a musical instrument}}\}
1.8
1.
n(P) = 0
2.
n(A) = 26
1.9
1.
finite
2.
infinite
1.10
1.
Set B is equal to set A, B = A
2.
neither
3.
Set B is equivalent to set C, B \sim C
1.11
1.
\{ {\text{heads, tails}}\} ;\{ {\text{heads}}\} ,{\text{ }}\{ {\text{tails}}\} ; and \emptyset
1.12
1.

A set with one member could contain any one of the following:

\{ {\text{Articuno}}\} {\text{, }}\{ {\text{Zapdos}}\} {\text{, }}\{ {\text{Moltres}}\} {\text{, or }}\{ {\text{Mewtwo}}\}.

2.

Any of the following combinations of three members would work:

\{ {\text{Articuno, Zapdos, Mewtwo}}\},\{ {\text{Articuno, Moltres, Mewtwo}}\}, or \{ {\text{Zapdos, Moltres, Mewtwo}}\}.

3.
The empty set is represented as \{ {\text{ }}\} or \emptyset .
1.13
1.
E \subset \mathbb{N}
1.14
1.
512
1.15
1.
\{ m|m = 5n{\text{ where }}n \in \mathbb{N}\}
1.16
1.

Serena also ordered a fish sandwich and chicken nuggets, because for the two sets to be equal they must contain the exact same items: {fish sandwich, chicken nuggets} = {fish sandwich, chicken nuggets}.

1.17
1.

There are multiple possible solutions. Each set must contain two players, but both players cannot be the same, otherwise the two sets would be equal, not equivalent. For example, {Maria, Shantelle} and {Angie, Maria}.

1.18
1.
The set of lions is a subset of the universal set of cats. In other words, the Venn diagram depicts the relationship that all lions are cats. This is expressed symbolically as {L} \subset {U}.
1.19
1.
The set of eagles and the set of canaries are two disjoint subsets of the universal set of all birds. No eagle is a canary, and no canary is an eagle.
1.20
1.
The universal set is the set of integers. Draw a rectangle and label it with U = \text{Integers}. Next, draw a circle in the rectangle and label with Natural numbers.
A Venn diagram shows a circle placed inside a rectangle. The circle represents natural numbers and is shaded in yellow. The rectangle represents U equals integers and is shaded in blue.
Venn diagram with universal set, U=\text{Integers}, and subset \mathbb{N} = {\text{Natural numbers}}.
2.
A Venn diagram shows a circle placed inside a rectangle. The circle represents A and is shaded in yellow. The rectangle represents U and is shaded in blue.
Venn Diagram with universal set, U and subset A.
1.21
1.
A Venn diagram shows two circles are placed inside a rectangle. The circle on the left represents birds and is shaded in orange. The circle on the right represents airplanes and is shaded in yellow. The rectangle represents U equals things that can fly and is shaded in blue.
Venn Diagram with universal set, U = Things that can fly with disjoint subsets Airplanes and Birds.
1.22
1.
A' = \left\{ {{\text{orange, green, indigo, violet}}} \right\}
2.
A' = \{ c \in U|c{\text{ is a lion}}\} or A' = \{ c \in U|c \notin A\}
1.23
1.
A\mathop \cap \nolimits B = \{ a\}
1.24
1.
A \cap B = \{ \,\,\}
1.25
1.
A \cap B = B = \{a,e,i,o,u\}
1.26
1.
A \cup B = \{a,d,h,p,s,y\}
1.27
1.
A \cup B = \{{\text{red, yellow, blue, orange, green, purple\}}}
1.28
1.
A \cup B = \{a, b, c,\ldots,z\} }} = A.
1.29
1.

33

1.30
1.

113

1.31
1.
A or B = A \cup B = \{h, a, p, y, w, e, s, o, m\} .
2.
A and C = A \cap C = \{a, h\} .
3.
B or C = B \cup C = \{a, w, e, s, o, m, t, h\} .
4.
(A and C) and B = (A \cap C) \cap B = \{a, h\} \cap \{a, w, e, s, o, m\} = \{a\}.
1.32
1.

127

2.

50

1.33
1.

A \cap B = \{ 3,5,7\}.

2.

A \cup B = \{ 1,2,3,5,7,9\}.

3.

A \cap B' = \{ 1,9\}.

4.

n(A \cap B') = 2.

1.34
1.
40
2.
0
3.
27
1.35
1.

n(B) = n(A{B^ + }) + n(A{B^ - }) + n({B^ + }) + n({B^ - }) = 14

2.

n({B^\prime }) = n(U) - n(B) = 86

3.

n(B \cup R{h^ + }) = 87

1.36
1.
A Venn diagram shows three intersecting circles inside a rectangle. The first circle representing soup is shaded in yellow and has a value of 4. The second circle representing the sandwich is shaded in red and has a value of 15. The third circle representing salad is shaded in blue and has a value of 8. The intersecting region of soup and sandwich has a value of 2. The intersecting region of soup and salad has a value of 3. The intersecting region of sandwich and salad has a value of 10. The intersecting region of all three circles has a value of 8. The rectangle represents U equals conference attendees equals 50 and has a value of 0.
Venn diagram – Attendees at a conference with sets: Soup, Sandwich, Salad – Complete Solution
1.37
1.
A \cap (B \cap C) = \{ 0,1,2,3,4,5,6\} \cap \{ 0,6,12\} = \{ 0,6\}
2.
(A \cap B) \cup (B \cap C) = \{ 0,2,4,6\} \cup \{ 0,3,6\} = \{ 0,2,3,4,6\}
3.
(A \cup {C^\prime }) \cap (B \cup {C^\prime }) = \{ 0,1,2,3,4,5,6,7,8,10,11\} \cap \{ 0,1,2,4,5,6,7,8,10,11,12\} = \{ 0,1,2,4,5,6,7,8,10,11\}
1.38
1.

The left side of the equation is:

Two Venn diagrams. The first diagram represents A intersection B. It shows two intersecting circles A and B placed inside a rectangle. The rectangle represents U. The intersecting region of the two circles is shaded in blue. The second diagram represents the complement of A intersection B. It shows two intersecting circles A and B placed inside a rectangle. The rectangle represents U. Except for the intersecting region, the other regions of the two circles are shaded in blue.
Venn diagram of intersection of two sets and its complement.

The right side of the equation is given by:

Three Venn diagrams. The first diagram represents A complement. A rectangle U with a circle A on its left. The region inside the rectangle, outside the circle, is shaded in blue. The second diagram represents a B complement. A rectangle U with a circle B on its right. The region inside the rectangle, outside the circle, is shaded in yellow. The third diagram represents A complement union B complement. Two intersecting circles A and B are placed inside a rectangle. The rectangle represents U. Except for the region of intersection, all other regions of the two circles are shaded in green.
Venn diagram of union of the complement of two sets.

Check Your Understanding

1.
set
2.
cardinality
3.
not a well-defined set
4.
12
5.
equivalent, but not equal
6.
finite
7.

Roster method: \{ \text{A,B,C,} \ldots \text{,Z}\}, and set builder notation: \{ x|x{\text{is a capital letter of the English alphabet}}\}

8.
subset
9.

To be a subset of a set, every member of the subset must also be a member of the set. To be a proper subset, there must be at least one member of the set that is not also in the subset.

10.
empty
11.
true
12.
{2^{10}} = 1024
13.
equivalent
14.
equal
15.
relationship
16.
universal
17.
disjoint or non-overlapping
18.
complement
19.
disjoint
20.

intersection

21.

union

22.

A \cup B

23.

A \cap B

24.
A
25.
B
26.
empty
27.
n(A \cup B) = n(A) + n(B) - n(A \cap B)
28.

overlap

29.

central

30.
intersection of all three sets, A \cap B \cap C
31.

parentheses, complement

32.
equation, true
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