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Contemporary Mathematics

Chapter Review

Contemporary MathematicsChapter Review
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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

Chapter Review

Statements and Quantifiers

Fill in the blanks to complete the following sentences.

1.
The ______________ of a logical statement has the opposite truth value of the original statement.
2.
_______________ are logical statements presented as the facts used to support the conclusion of a logical argument.
Determine whether each of the following sentences represents a logical statement, also called a proposition. If it is a logical statement, determine whether it is true or false.
3.
Where is the restroom?
4.
No even numbers are odd numbers.
5.
4 + 3 = 8.
Write the negation of each following statement symbolically and in words.
6.
\text{~}p{:} Pink Floyd’s album The Wall is not a rock opera.
7.
q{:} Some dogs are Labrador retrievers.
8.
\text{~}r{:} Some universities are not expensive.
Draw a logical conclusion to the following arguments, and include in both one of the following quantifiers: all, some, or none.
9.
Spaghetti noodles are made with wheat, ramen noodles are made with wheat, and lo mein noodles are made with wheat.
10.
A Porsche Boxster does not have four doors, a Volkswagen Beetle does not have four doors, and a Mazda Miata does not have four doors.
Compound Statements

Fill in the blanks to complete the following sentences.

11.
___________________ are words or symbols used to join two or more logical statement together to from a compound statement.
12.
__________________ and __________________ have equal dominance and are evaluated from left to right when no parentheses are present in a compound logical statement.
Translate each compound statement below into symbolic form.
Given: p{:} “Tweety Bird is a bird,” q{:} “Bugs is a bunny,” r{:} “Bugs says, ‘What’s up, Doc?’,” s{:} “Sylvester is a cat,” and t{:} “Sylvester chases Tweety Bird.”
13.
If Tweety Bird is a bird, then Sylvester will not chase him.
14.
Tweety Bird is a bird and Sylvester chases him if and only if Bugs says, “What’s up Doc?”
Translate the symbolic form of each compound logical statement below into words.

Given: p{:} “Tweety Bird is a bird,” q{:} “Bugs is a bunny,” r{:} “Bugs says, ‘What’s up, Doc?’,” s{:} “Sylvester is a cat,” and t{:} “Sylvester chases Tweety Bird.”

15.
\text{~}q \vee p \to \text{~}s
16.
\text{~}(p \wedge \text{~}s) \leftrightarrow q \to r

For each of the following compound logical statements, apply the proper dominance of connectives by adding parentheses to indicate the order to evaluate the statement.

17.
p \vee q \wedge r \to {\text{ }}\text{~}s \wedge t
18.
\text{~}p \to q \vee r \leftrightarrow p \wedge s \to \text{~}t
Constructing Truth Tables

Fill in the blanks to complete the sentences.

19.
A ______________ is true if at least one of its component statements is true.
20.
For a ____________________ to be true, all of its component statements must be true.
Given the statements, p{:} “No fish are mammals,” q{:} “All lions are cats,” and \text{~}r{:} “Some birds do not lay eggs,” construct a truth table to determine the truth value of each compound statement below.
21.
p \wedge \text{~}r
22.
\text{~}\left( {p \vee \text{~}r} \right)
23.
\text{~}p \vee q \wedge \text{~}\left( {\text{~}r} \right)
Construct a truth table to analyze all the possible outcomes of the following statements, and determine whether the statements are valid.
24.
\text{~}p \vee q \wedge p
25.
\text{~}p \vee q \vee \text{~}q
Truth Tables for the Conditional and Biconditional

Fill in the blanks to complete the following sentences.

26.
If the ________, p, of a conditional statement is true, then the conclusion, q, must also be true for the conditional statement p \to q to be true.
27.
The biconditional statement p \leftrightarrow q is __________ whenever the truth value of p matches the truth value of q, otherwise it is _________.

Complete the truth tables below to determine the truth value of the proposition in the last column.

28.
p q r {p} \vee {q} \text{~}\left( {{p} \vee {q}} \right) \text{~}{r} \text{~}({p} \vee {q}) \to {\text{ }}\text{~}{r}
F F T
29.
p q \text{~}{q} {p} \to {q} \text{~}\left( {{p} \to {q}} \right) {p} \wedge \text{~}{q} \text{~}({p} \to {q}) \leftrightarrow \left( {{p} \wedge \text{~}{q}} \right)
T F
Assume the following statements are true. p{:} “Poof is a baby fairy,” q{:} “Timmy Turner has fairly odd parents,” r{:} “Cosmo and Wanda will grant Timmy’s wishes,” and t{:} “Timmy Turner is 10 years old.” Translate each of the following statements into symbolic form, then determine its truth value.
30.
If Timmy Turner is 10 years old and Poof is not a baby fairy, then Timmy Turner has fairly odd parents.
31.
Cosmos and Wanda will not grant Timmy’s wishes if and only if Timmy Turner is 10 years old or he does not have fairly odd parents.
32.
Construct a truth table to analyze all the possible outcomes and determine the validity of the following argument.

\text{~}p \vee q{\text{ }} \leftrightarrow {\text{ }}\text{~}q \to {\text{ }}\text{~}p

Equivalent Statements
Fill in the blanks to complete the sentences below.
33.
The _________________ is logically equivalent to the inverse \text{~}p \to\text{~}q.
34.
The _________________ is logically equivalent to the conditional p \to q.
Use the conditional statement, p \to q{:} “If Novak makes the basket, then Novak’s team will win the game," to answer the following questions.
35.
Write the conclusion of the conditional statement in words and label it appropriately.
36.
Write the hypothesis of the conditional statement in words and label it appropriately.
37.
Identify the following statement as the converse, inverse, or contrapositive: “If Novak does not make the basket, then his team will not win the game.”
38.
Identify the following statement as the converse, inverse, or contrapositive: “If Novak’s team wins the game, then he made the basket.”
De Morgan’s Laws
Fill in the blanks to complete the sentences.
39.
De Morgan’s Law for the negation of a disjunction states that \text{~}(p \vee q) \equiv ____________.
40.
De Morgan’s Law for the negation of a conjunctions states that _____________ \equiv {\text{ }}\text{~}p{\text{ }} \vee \text{~}q.
41.
Apply De Morgan’s Law to write the statement without parentheses: .\text{~}\left( {\text{~}p \wedge q} \right).
42.
Apply the property for the negation of a conditional to write the statement as a conjunction or disjunction: .\text{~}(\text{~}p \wedge q \to\text{~}r).
43.
Write the negation of the conditional statement in words: If Thomas Edison invented the phonograph, then albums are made of vinyl, or the transistor radio was the first portable music device.
44.
Construct a truth table to verify that the logical property is valid: \text{~}\left( {\text{~}p \to\text{~}q} \right) \equiv \text{~}p \wedge q.
Logical Arguments

Fill in the blanks to complete the sentences below.

45.
The _____ ___ __________ is a valid logical argument with premises, p \to q and p, used to support the conclusion, q.
46.
The chain rule for conditional arguments states that the ___________________ property applies to conditional arguments, so that: \left( {p \to q} \right) \wedge \left( {q \to r} \right) \to \left( {p \to r} \right).
Assume each pair of statements represents true premises in a logical argument. Based on these premises, state a valid conclusion that is consistent with the form of the argument.
47.
If the Tampa Bay Buccaneers did not win Super Bowl LV, then Tom Brady was not their quarterback. Tom Brady was the Tampa Bay Buccaneers quarterback.
48.
If \text{~}q, then p and if r, then \text{~}q.
49.
If Kamala Harris is the vice president of the United States, then Kamala Harris is the president of the U.S. Senate. Kamala Harris is the vice president of the United States.
50.
Construct a truth table or Venn diagram to prove whether the following argument is valid. If the argument is valid, determine whether it is sound.
If all frogs are brown, then Kermit is not a frog. Kermit is a frog. Therefore, some frogs are not brown.
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