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Contemporary Mathematics

Chapter 2

Contemporary MathematicsChapter 2
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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

2.1
1.
Logical statement, false.
2.
Logical statement, true.
3.
Not a logical statement, questions cannot be determined to be either true or false.
2.2
1.
p{\text{:}} The movie Gandhi won the Academy Award for Best Picture in 1982.
2.
q{\text{:}} Soccer is the most popular sport in the world.
3.
r{\text{:}} All oranges are citrus fruits.
2.3
1.
Ted Cruz was born in Texas.
2.
Adele does not have a beautiful voice.
3.
Leaves do not convert carbon dioxide to oxygen during the process of photosynthesis.
2.4
1.
p
2.
\text~q
3.
\text~r
2.5
1.
p
2.
Woody and Buzz Lightyear are not best friends.
2.6
1.
The sum of some consecutive integers results in a prime number.
2.

No birds give live birth to their young.

3.

All squares are parallelograms and have four sides.

2.7
1.
All apples are sweet.
2.
Some triangles are squares.
3.
No vegetables are green.
2.8
1.
Negation; \sim.
2.
Conjunction; \wedge.
3.
Biconditional; \leftrightarrow.
2.9
1.
q \wedge \text{~}p
2.
q \leftrightarrow p
3.
p \vee \text{~}q
4.
p \to q
2.10
1.
If our friends did not come over to watch the game, then my roommates ordered pizza or I ordered wings.
2.
If my roommates ordered pizza and I ordered wings, then our friends came over to watch the game.
3.
It is not the case that my roommates ordered pizza or our friends came over to watch the game.
2.11
1.
((p \vee q) \wedge (\text{~}r))
2.
((\text{~}p) \to (q \vee r))
3.
((\text{~}p) \vee (\text{~}q)) \leftrightarrow (\text{~}(p \wedge q)); this is another example of De Morgan’s Laws and it is always true.
2.12
1.
p{\text{:}}\,3 \times 5 \ne 14, true
2.
q: No houses are built with bricks; false
3.
\text~r: Abuja is not the capital of Nigeria; false
2.13
1.
True
2.
False
3.
True
2.14
1.
True
2.
False
3.
True
2.15
1.
p q r \text{~}q \text{~}q \wedge p (\text{~}q \wedge p) \vee r
T T F F F F
false
2.
p q r \text{~}r p \vee q (p \vee q) \wedge (\text{~}r)
T T F T T T
true
3.
p q r (p \wedge r) \text{~}(p \wedge r) \text{~}(p \wedge r) \wedge q
T T F F T T
true
2.16
1.
p q \text~q p \wedge \text{~}q
T T F F
T F T T
F T F F
F F T F
2.
p q p \vee q \text{~}(p \vee q)
T T T F
T F T F
F T T F
F F F T
3.
p q r \text{~}q p\, \wedge \text{~}q (p \wedge \text{~}q) \vee r
T T T F F T
T T F F F F
T F T T T T
T F F T T T
F T T F F T
F T F F F F
F F T T F T
F F F T F F
2.17
1.
Valid
p \text{~}{p} {p}\vee \text{~}{p}
T F T
F T T
2.
Not valid
p q \text{~}{p} \text{~}{q} \~p \vee ~q
T T F F F
T F F T T
F T T F T
F F T T T
2.18
1.
False
p q {p} \to {q}
T F F
2.
True
p q \text{~}{q} {p} \to \text{~}{q}
T F T T
3.
True
p q \text{~}{p} \text{~}{p} \to {q}
T F F T
2.19
1.
Valid
p q \text{~}{p} \text{~}{p} \vee {q} {q} \to \left( {\text{~}{p} \vee {q}} \right)
T T F T T
T F F F T
F T T T T
F F T T T
2.
Not valid
p q \text{~}{p} {q} \wedge {p} \text{~}{p} \to \left( {{q} \wedge {p}} \right)
T T F T T
T F F F T
F T T F F
F F T F F
2.20
1.
False
p q {p} \leftrightarrow {q}
T F F
2.
True
p q \text{~}{q} {p} \leftrightarrow \text{~}{q}
T F T T
3.
True
p q \text{~}{p} \text{~}{p} \leftrightarrow {q}
T F F T
2.21
1.
Valid
p q {p} \wedge {q} \text{~}\left( {{p} \wedge {q}} \right) \text{~}{p} \text{~}{q} \text{~}{p} \vee \text{~}{q} \text{~}\left( {{p} \wedge {q}} \right) \leftrightarrow \left( {\text{~}{p} \vee \text{~}{q}} \right)
T T T F F F F T
T F F T F T T T
F T F T T F T T
F F F T T T T T
2.
Not Valid
p q \text{~}{p} {q} \wedge {p} \text{~}{p} \leftrightarrow \left( {{q} \wedge {p}} \right)
T T F T F
T F F F T
F T T F F
F F T F F
3.
Valid
p q {p} \to {q} \text{~}{p} \text{~}{p} \vee {q} \left( {p \to q} \right) \leftrightarrow \left( {\text{~}p \vee q} \right)
T T T F T T
T F F F F T
F T T T T T
F F T T T T
4.
Valid
p q r \text{~}{p} \text{~}{q} {p} \wedge {q} \left( {{p} \wedge {q}} \right) \to {r} \text{~}{p} \vee \text{~}{q} \left( {{\text{~}p} \vee {\text{~}q}} \right) \vee {r} \left( {{p} \wedge {q} \to {r}} \right) \leftrightarrow \left( {{\text{~}p} \vee {\text{~}q} \vee {r}} \right)
T T T F F T T F T T
T T F F F T F F F T
T F T F T F T T T T
T F F F T F T T T T
F T T T F F T T T T
F T F T F F T T T T
F F T T T F T T T T
F F F T T F T T T T
2.22
1.
p \to q is logically equivalent to \text{~}q \to\text{~}p.
p {q} {p} \to {q} \text{~}{q} \text{~}{p} \text{~}{q} \to \text{~}{p} \left( {p} \to {q} \right) \leftrightarrow \left( {\text{~}{q} \to \text{~}{p}} \right)
T T T F F T T
T F F T F F T
F T T F T T T
F F T T T T T
2.
p \to q is not logically equivalent to p \vee \text{~}q.
p q {p} \to {q} \text{~}{q} {p}\rm \vee \text{~}{q} \left( {{p} \to {q}} \right) \leftrightarrow \left( {{p}{\rm{ }} \vee \text{~}{q}} \right)
T T T F T T
T F F T T F
F T T F F F
F F T T T T
2.23
1.

If Elvis Presley wore capes, then some superheroes wear capes.

2.

If some superheroes wear capes, then Elvis Presley wore capes.

3.

If Elvis Presley did not wear capes, then no superheroes wear capes.

4.

If no superheroes wear capes, then Elvis Presley did not wear capes.

2.24
1.
p: Dora is an explorer.

2.
q: Boots is a monkey.
3.
Inverse
4.
Converse
5.
Converse
2.25
1.
If my friend does not live in California, then my friend lives in San Francisco. True.
2.
If my friend does not live in San Francisco, then my friend lives in California. True.
3.
If my friend lives in California, then my friend does not live in San Francisco. False.
2.26
1.
Jackie did not play softball and she did not run track.
2.

Brandon did not study for his certification exam, or he did not pass his exam.

2.27
1.
Edna Mode made a new superhero costume, and it includes a cape.
2.

I had pancakes for breakfast, and I did not use maple syrup.

2.28
1.
Some people like ice cream, but ice cream makers will not make a profit.
2.
Raquel cannot play video games, but somebody will play video games.
2.29
1.
Eric needs to replace the light bulb, and Marcos did not leave the light bulb on all night, and Dan did not break the light bulb.
2.
Trenton went to school, and Regina went to work, and Merika did not clean the house.
2.30
1.
p q {p} \vee {q} \text{~}\left( {{p} \vee {q}} \right) \text{~}p \text{~}q \text{~}p \wedge \text{~}q \text{~}({p} \vee {q}) \leftrightarrow \left( {\text{~}p \wedge \text{~}q} \right)
T T T F F F F T
T F T F F T F T
F T T F T F F T
F F F T T T T T
2.31
1.
Some people like history.
2.

Some people do not like reading.

3.

The polygon is not an octagon.

2.32
1.
My classmate does not like history.
2.
Homer likes to read.
3.
The polygon does not have five sides.
2.33
1.
If my roommate does not go to work, then they will not be able to pay their bills.
2.
If penguins cannot fly, then we will watch the news.
3.
If Marcy goes to the movies, then she will buy water.

Check Your Understanding

1.
logical statement
2.

negation

3.
\text{~}p
4.

p

5.

premises

6.

Inductive

7.

quantifiers

8.
Some giraffes are not tall.
9.
compound statement
10.

connective

11.

biconditional, \leftrightarrow

12.

Parentheses, (\,)

13.

Conjunction, \wedge; disjunction, \vee (in any order)

14.

valid

15.

true

16.

truth table

17.

four

18.

two

19.
one-way contract
20.
conclusion
21.
hypothesis
22.
biconditional
23.
biconditional
24.
true
25.
always true, valid, or a tautology.
26.
conditional
27.
logically equivalent
28.
inverse
29.
converse, inverse
30.
\text{~}p \vee \text{~}q
31.
\text{~}p \wedge \text{~}q
32.
p \wedge \text{~}q
33.
De Morgan’s Laws
34.
premise
35.
valid
36.
inductive
37.
deductive
38.
fallacy
39.
sound
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