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Contemporary Mathematics

2.5 Equivalent Statements

Contemporary Mathematics2.5 Equivalent Statements
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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
Two people are standing outside. One speaks with his hands open, while the other listens.
Figure 2.11 How your logical argument is stated affects the response, just like how you speak when holding a conversation can affect how your words are received. (credit: modification of work by Goelshivi/Flickr, Public Domain Mark 1.0)

Learning Objectives

After completing this section, you should be able to:

  1. Determine whether two statements are logically equivalent using a truth table.
  2. Compose the converse, inverse, and contrapositive of a conditional statement

Have you ever had a conversation with or sent a note to someone, only to have them misunderstand what you intended to convey? The way you choose to express your ideas can be as, or even more, important than what you are saying. If your goal is to convince someone that what you are saying is correct, you will not want to alienate them by choosing your words poorly.

Logical arguments can be stated in many different ways that still ultimately result in the same valid conclusion. Part of the art of constructing a persuasive argument is knowing how to arrange the facts and conclusion to elicit the desired response from the intended audience.

In this section, you will learn how to determine whether two statements are logically equivalent using truth tables, and then you will apply this knowledge to compose logically equivalent forms of the conditional statement. Developing this skill will provide the additional skills and knowledge needed to construct well-reasoned, persuasive arguments that can be customized to address specific audiences.

Checkpoint

An alternate way to think about logical equivalence is that the truth values have to match. That is, whenever pp is true, qq is also true, and whenever pp is false, qq is also false.

Determine Logical Equivalence

Two statements, pp and qq, are logically equivalent when p↔qp↔q is a valid argument, or when the last column of the truth table consists of only true values. When a logical statement is always true, it is known as a tautology. To determine whether two statements pp and qq are logically equivalent, construct a truth table for p↔qp↔q and determine whether it valid. If the last column is all true, the argument is a tautology, it is valid, and pp is logically equivalent to qq; otherwise, pp is not logically equivalent to qq.

Example 2.22

Determining Logical Equivalence with a Truth Table

Create a truth table to determine whether the following compound statements are logically equivalent.

  1. p→q;p→q; ~p→ ~q~p→ ~q
  2. p→q;p→q; ~p∨q~p∨q

Your Turn 2.22

Create a truth table to determine whether the following compound statements are logically equivalent.
1.
p \to q; ~q \to \text~p
2.
p \to q;\, p \vee \text{~}q

Compose the Converse, Inverse, and Contrapositive of a Conditional Statement

The converse, inverse, and contrapositive are variations of the conditional statement, p→q.p→q.

  • The converse is if qq then pp, and it is formed by interchanging the hypothesis and the conclusion. The converse is logically equivalent to the inverse.
  • The inverse is if ~p~p then ~q~q, and it is formed by negating both the hypothesis and the conclusion. The inverse is logically equivalent to the converse.
  • The contrapositive is if ~q~q then ~p~p, and it is formed by interchanging and negating both the hypothesis and the conclusion. The contrapositive is logically equivalent to the conditional.

The table below shows how these variations are presented symbolically.

Conditional Contrapositive Converse Inverse
pp qq ~p~p ~q~q p→qp→q ~q→ ~p~q→ ~p q→pq→p ~p→ ~q~p→ ~q
T T F F T T T T
T F F T F F T T
F T T F T T F F
F F T T T T T T

Example 2.23

Writing the Converse, Inverse, and Contrapositive of a Conditional Statement

Use the statements, pp: Harry is a wizard and qq: Hermione is a witch, to write the following statements:

  1. Write the conditional statement, p→qp→q, in words.
  2. Write the converse statement, q→pq→p, in words.
  3. Write the inverse statement, ~p→ ~q~p→ ~q, in words.
  4. Write the contrapositive statement, ~q→ ~p~q→ ~p, in words.

Your Turn 2.23

Use the statements, p: Elvis Presley wore capes and q: Some superheroes wear capes, to write the following statements:
1.
Write the conditional statement, p \to q, in words.
2.
Write the converse statement, q \to p, in words.
3.
Write the inverse statement, \text{~}p \to {\rm{ }}\text{~}q, in words.
4.
Write the contrapositive statement, \text{~}q \to {\rm{ }}\text{~}p, in words.

Example 2.24

Identifying the Converse, Inverse, and Contrapositive

Use the conditional statement, “If all dogs bark, then Lassie likes to bark,” to identify the following.

  1. Write the hypothesis of the conditional statement and label it with a pp.
  2. Write the conclusion of the conditional statement and label it with a qq.
  3. Identify the following statement as the converse, inverse, or contrapositive: “If Lassie likes to bark, then all dogs bark.”
  4. Identify the following statement as the converse, inverse, or contrapositive: “If Lassie does not like to bark, then some dogs do not bark.”
  5. Which statement is logically equivalent to the conditional statement?

Your Turn 2.24

Use the conditional statement, “If Dora is an explorer, then Boots is a monkey,” to identify the following:
1.
Write the hypothesis of the conditional statement and label it with a p.
2.
Write the conclusion of the conditional statement and label it with a q.
3.
Identify the following statement as the converse, inverse, or contrapositive: “If Dora is not an explorer, then Boots is not a monkey.”
4.
Identify the following statement as the converse, inverse, or contrapositive: “If Boots is a monkey, then Dora is an explorer.”
5.
Which statement is logically equivalent to the inverse?

Example 2.25

Determining the Truth Value of the Converse, Inverse, and Contrapositive

Assume the conditional statement, p→q:p→q: “If Chadwick Boseman was an actor, then Chadwick Boseman did not star in the movie Black Panther” is false, and use it to answer the following questions.

  1. Write the converse of the statement in words and determine its truth value.
  2. Write the inverse of the statement in words and determine its truth value.
  3. Write the contrapositive of the statement in words and determine its truth value.

Your Turn 2.25

Assume the conditional statement p \to q: “If my friend lives in San Francisco, then my friend does not live in California” is false, and use it to answer the following questions.
1.
Write the converse of the statement in words and determine its truth value.
2.
Write the inverse of the statement in words and determine its truth value.
3.
Write the contrapositive of the statement in words and determine its truth value.

Check Your Understanding

25.
Two statements p and q are logically equivalent to each other if the biconditional statement, p \leftrightarrow q is ________________.
26.
The _____ statement has the form, “p then q.”
27.
The contrapositive is _____________ ___________ to the conditional statement, and has the form, "if \text~q, then \text~p."
28.
The _________________ of the conditional statement has the form, "if \text~p, then \text~q."
29.
The _________________ of the conditional statement is logically equivalent to the _______________ and has the form, "if q then p."

Section 2.5 Exercises

For the following exercises, determine whether the pair of compound statements are logically equivalent by constructing a truth table.
1.
Converse: q \to p and inverse: \text{~}p \to {\rm{ }}\text{~}q
2.
Conditional: p \to q and contrapositive: \text{~}q \to {\rm{ }}\text{~}p
3.
Inverse: \text{~}p \to {\rm{ }}\text{~}q and contrapositive: \text{~}q \to {\rm{ }}\text{~}p
4.
Conditional: p \to q and converse: q \to p
5.
\text{~}p \to q and p \vee \text{~}q
6.
\text{~}p \to q and p \vee q
7.
\text{~}(p \wedge q) and \text{~}p \wedge \text{~}q
8.
\text{~}(p \wedge q) and \text{~}p \vee \text{~}q
9.
p \wedge (q \vee r) and \left( {p \wedge q} \right) \vee \left( {p \wedge r} \right)
10.
p \wedge \left( {q \vee r} \right) and \left( {p \wedge q} \right) \vee r
For the following exercises, answer the following:
  1. Write the conditional statement p \to q in words.
  2. Write the converse statement q \to p in words.
  3. Write the inverse statement \text{~}p \to \text{~}q in words.
  4. Write the contrapositive statement \text{~}q \to \text{~}p in words.
11.
p: Six is afraid of Seven and q: Seven ate Nine.
12.
p: Hope is eternal and q: Despair is temporary.
13.
p: Tom Brady is a quarterback and q: Tom Brady does not play soccer.
14.
p: Shakira does not sing opera and q: Shakira sings popular music.
15.
p:The shape does not have three sides and q: The shape is not a triangle.
16.
p: All birds can fly and q: Emus can fly.
17.
p: Penguins cannot fly and q: Some birds can fly.
18.
p: Some superheroes do not wear capes and q: Spiderman is a superhero.
19.
p: No Pokémon are little ponies and q: Bulbasaur is a Pokémon.
20.
p: Roses are red, and violets are blue and q: Sugar is sweet, and you are sweet too.
For the following exercises,use the conditional statement: “If Clark Kent is Superman, then Lois Lane is not a reporter,” to answer the following questions.
21.
Write the hypothesis of the conditional statement, label it with a p, and determine its truth value.
22.
Write the conclusion of the conditional statement, label it with a , and determine its truth value.
23.
Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: “If Clark Kent is not Superman, then Lois Lane is a reporter.”
24.
Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: “If Lois Lane is a reporter, then Clark Kent is not Superman.”
25.
Which form of the conditional is logically equivalent to the converse?
For the following exercises, use the conditional statement: “If The Masked Singer is not a music competition, then Donnie Wahlberg was a member of New Kids on the Block,” to answer the following questions.
26.
Write the hypothesis of the conditional statement, label it with a p, and determine its truth value.
27.
Write the conclusion of the conditional statement, label it with a q, and determine its truth value.
28.
Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: “If Donnie Wahlberg was a member of New Kids on the Block, then The Masked Singer is not a music competition.”
29.
Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: “If The Masked Singer is a music competition, then Donnie Wahlberg was not a member of New Kids on the Block.”
30.
Which form of the conditional is logically equivalent to the contrapositive, \text{~}q \to {\text{~}p?
For the following exercises, use the conditional statement: “If all whales are mammals, then no fish are whales,” to answer the following questions.
31.
Write the hypothesis of the conditional statement, label it with a p, and determine its truth value.
32.
Write the conclusion of the conditional statement, label it with a q, and determine its truth value.
33.
Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: “If some fish are whales, then some whales are not mammals.”
34.
Write the inverse in words and determine its truth value.
35.
Write the converse in words and determine its truth value.
For the following exercises, use the conditional statement: “If some parallelograms are rectangles, then some circles are not symmetrical,” to answer the following questions.
36.
Write the hypothesis of the conditional statement, label it with a p, and determine its truth value.
37.
Write the conclusion of the conditional statement, label it with a q, and determine its truth value.
38.
Write the converse in words and determine its truth value.
39.
Write the contrapositive in words and determine its truth value.
40.
Write the inverse in words and determine its truth value.
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