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Finite Mathematics and Calculus with Applications

Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Chapter 6

Logic

Educators


Problem 1

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
Montevideo is the capital of Uruguay.

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Problem 2

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
John Jay was the first chief justice of the United States.

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Problem 3

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
Don't feed the animals.

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Problem 4

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
Do unto others as you would have them do unto you.

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Problem 5

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
$2+2=5$ and $3+3=7$

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Problem 6

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
$x < 7$ or $x > 14$

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Problem 7

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
Got milk?

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Problem 8

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
Is that all there is?

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Problem 9

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
I am not a crook.

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Problem 10

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
China does not have a population of more than 1 billion.

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Problem 11

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
She enjoyed the comedy team of Penn and Teller.

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Problem 12

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
The New Hampshire motto is "Live free or die."

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Problem 13

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
If I get an A.I will celebrate.

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Problem 14

Decide whether each of the following is a statement. If it is a statement, decide whether or not it is compound.
If it's past $8 : 00,$ then we are late.

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Problem 15

Write a negation for each statement.
My favorite flavor is chocolate.

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Problem 16

Write a negation for each statement.
This is not the time to complain.

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Problem 17

Give a negation of each inequality.
$$
y>12
$$

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Problem 18

Give a negation of each inequality.
$$
x<-6
$$

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Problem 19

Give a negation of each inequality.
$$
q \geq 5
$$

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Problem 20

Give a negation of each inequality.
$$
r \leq 19
$$

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Problem 21

Try to negate the sentence "The exact number of words in this sentence is $10^{\prime \prime}$ and see what happens. Explain the problem that arises.

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Problem 22

Explain why the negation of $" r>4 "$ is not $^{4} r<4 . "$

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Problem 23

Let $b$ represent the statement "I'm getting better" and $d$ represent the statement "My parrot is dead." Translate each symbolic statement into words.
$\sim b$

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Problem 24

Let $b$ represent the statement "I'm getting better" and $d$ represent the statement "My parrot is dead." Translate each symbolic statement into words.
$-d$

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Problem 25

Let $b$ represent the statement "I'm getting better" and $d$ represent the statement "My parrot is dead." Translate each symbolic statement into words.
$\sim b \vee d$

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Problem 26

Let $b$ represent the statement "I'm getting better" and $d$ represent the statement "My parrot is dead." Translate each symbolic statement into words.
$b \wedge \sim d$

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Problem 27

Let $b$ represent the statement "I'm getting better" and $d$ represent the statement "My parrot is dead." Translate each symbolic statement into words.
$\sim(b \wedge \sim d)$

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Problem 28

Let $b$ represent the statement "I'm getting better" and $d$ represent the statement "My parrot is dead." Translate each symbolic statement into words.
$\sim(b \vee d)$

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Problem 29

Use the concepts introduced in this section to answer Exercises $29-34 .$
If $q$ is false, what must be the truth value of $(p \wedge \sim q) \wedge q ?$

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Problem 30

Use the concepts introduced in this section to answer Exercises $29-34 .$
If $q$ is true, what must be the truth value of $q \vee(q \wedge \sim p) ?$

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Problem 31

Use the concepts introduced in this section to answer Exercises $29-34 .$
If $p \wedge q$ is true, then $q$ must be

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Problem 32

Use the concepts introduced in this section to answer Exercises $29-34 .$
If $p \vee q$ is false, and $p$ is false, then $q$ must be _______

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Problem 33

Use the concepts introduced in this section to answer Exercises $29-34 .$
If $\sim(p \vee q)$ is true, what must be the truth values of each of the component statements?

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Problem 34

Use the concepts introduced in this section to answer Exercises $29-34 .$
If $-(p \wedge q)$ is false, what must be the truth values of the component statements?

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Problem 35

Let $p$ represent a false statement and let $q$ represent a true statement. Find the truth value of each compound statement.
$\sim p$

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Problem 36

Let $p$ represent a false statement and let $q$ represent a true statement. Find the truth value of each compound statement.
$\sim q$

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Problem 37

Let $p$ represent a false statement and let $q$ represent a true statement. Find the truth value of each compound statement.
$p \vee q$

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Problem 38

Let $p$ represent a false statement and let $q$ represent a true statement. Find the truth value of each compound statement.
$. p \wedge q$

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Problem 39

Let $p$ represent a false statement and let $q$ represent a true statement. Find the truth value of each compound statement.
$p \vee \sim q$

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Problem 40

Let $p$ represent a false statement and let $q$ represent a true statement. Find the truth value of each compound statement.
$\sim p \wedge q$

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Problem 41

Let $p$ represent a false statement and let $q$ represent a true statement. Find the truth value of each compound statement.
$\sim p \vee \sim q$

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Problem 42

Let $p$ represent a false statement and let $q$ represent a true statement. Find the truth value of each compound statement.
$p \wedge \sim q$

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Problem 43

Let $p$ represent a false statement and let $q$ represent a true statement. Find the truth value of each compound statement.
$\sim(p \wedge \sim q)$

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Problem 44

Let $p$ represent a false statement and let $q$ represent a true statement. Find the truth value of each compound statement.
$\sim(\sim p \vee \sim q)$

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Problem 45

Let $p$ represent a false statement and let $q$ represent a true statement. Find the truth value of each compound statement.
$\sim[\sim p \wedge(\sim q \vee p)]$

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Problem 46

Let $p$ represent a false statement and let $q$ represent a true statement. Find the truth value of each compound statement.
$\sim[(\sim p \wedge \sim q) \vee \sim q]$

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Problem 47

Is the statement $3 \geqq 1$ a conjunction or a disjunction? Why?

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Problem 48

Why is the statement $6 \geq 2$ true? Why is $6 \geq 6$ true?

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Problem 49

Let $p$ represent a true statement, and $q$ and $r$ represent false statements. Find the truth value of each compound statement.
$$
(p \wedge r) \vee \sim q
$$

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Problem 50

Let $p$ represent a true statement, and $q$ and $r$ represent false statements. Find the truth value of each compound statement.
$$
(q \vee \sim r) \wedge p
$$

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Problem 51

Let $p$ represent a true statement, and $q$ and $r$ represent false statements. Find the truth value of each compound statement.
$$
p \wedge(q \vee r)
$$

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Problem 52

Let $p$ represent a true statement, and $q$ and $r$ represent false statements. Find the truth value of each compound statement.
$$
(\sim p \wedge q) \vee \sim r
$$

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Problem 53

Let $p$ represent a true statement, and $q$ and $r$ represent false statements. Find the truth value of each compound statement.
$$
\sim(p \wedge q) \wedge(r \vee \sim q)
$$

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Problem 54

Let $p$ represent a true statement, and $q$ and $r$ represent false statements. Find the truth value of each compound statement.
$$
(\sim p \wedge \sim q) \vee(\sim r \wedge q)
$$

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Problem 55

Let $p$ represent a true statement, and $q$ and $r$ represent false statements. Find the truth value of each compound statement.
$$
\sim[(\sim p \wedge q) \vee r]
$$

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Problem 56

Let $p$ represent a true statement, and $q$ and $r$ represent false statements. Find the truth value of each compound statement.
$$
\sim[r \vee(\sim q \wedge \sim p)]
$$

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Problem 57

Let $p$ represent the statement $" 2 > 7,$ " let $q$ represent the statement $" 8 \leq 6, "$ and let $r$ represent the statement $^{6} 19 \leq 19^{39}$ Find the truth value of each compound statement.
$$
p \wedge r
$$

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Problem 58

Let $p$ represent the statement $" 2 > 7,$ " let $q$ represent the statement $" 8 \leq 6, "$ and let $r$ represent the statement $^{6} 19 \leq 19^{39}$ Find the truth value of each compound statement.
$$
p \vee \sim q
$$

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Problem 59

Let $p$ represent the statement $" 2 > 7,$ " let $q$ represent the statement $" 8 \leq 6, "$ and let $r$ represent the statement $^{6} 19 \leq 19^{39}$ Find the truth value of each compound statement.
$$
\sim q \vee \sim r
$$

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Problem 60

Let $p$ represent the statement $" 2 > 7,$ " let $q$ represent the statement $" 8 \leq 6, "$ and let $r$ represent the statement $^{6} 19 \leq 19^{39}$ Find the truth value of each compound statement.
$$
\sim p \wedge \sim r
$$

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Problem 61

Let $p$ represent the statement $" 2 > 7,$ " let $q$ represent the statement $" 8 \leq 6, "$ and let $r$ represent the statement $^{6} 19 \leq 19^{39}$ Find the truth value of each compound statement.
$$
(p \wedge q) \vee r
$$

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Problem 62

Let $p$ represent the statement $" 2 > 7,$ " let $q$ represent the statement $" 8 \leq 6, "$ and let $r$ represent the statement $^{6} 19 \leq 19^{39}$ Find the truth value of each compound statement.
$$
\sim p \vee(\sim r \vee \sim q)
$$

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Problem 63

Let $p$ represent the statement $" 2 > 7,$ " let $q$ represent the statement $" 8 \leq 6, "$ and let $r$ represent the statement $^{6} 19 \leq 19^{39}$ Find the truth value of each compound statement.
$$
(\sim r \wedge q) \vee \sim p
$$

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Problem 64

Let $p$ represent the statement $" 2 > 7,$ " let $q$ represent the statement $" 8 \leq 6, "$ and let $r$ represent the statement $^{6} 19 \leq 19^{39}$ Find the truth value of each compound statement.
$$
\sim(p \vee \sim q) \vee \sim r
$$

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Problem 65

Income Tax The following excerpts appear in a guide for preparing income tax reports. Source: Your Income Tax 2010 .
Which of these excerpts are statements?

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Problem 66

Income Tax The following excerpts appear in a guide for preparing income tax reports. Source: Your Income Tax 2010 .
Which of these excerpts are compound statements?

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Problem 67

Income Tax The following excerpts appear in a guide for preparing income tax reports. Source: Your Income Tax 2010 .
Write the negation of excerpt d.

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Problem 68

Income Tax The following excerpts appear in a guide for preparing income tax reports. Source: Your Income Tax 2010 .
Determine $p$ and $q$ to symbolically represent excerpt c.

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Problem 69

Technology For Exercises $69-74$ , let $a$ represent the statement "Apple Computer, Inc. developed the iPad," and $j$ represent the statement, "Steve Jobs is the CEO of Apple Computer, Inc." Convert each compound statement into symbols.
Apple Computer, Inc. developed the iPad, and Steve Jobs is the CEO of Apple Computer, Inc.

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Problem 70

Technology For Exercises $69-74$ , let $a$ represent the statement "Apple Computer, Inc. developed the iPad," and $j$ represent the statement, "Steve Jobs is the CEO of Apple Computer, Inc." Convert each compound statement into symbols.
Apple Computer, Inc. did not develop the iPad, and Steve Jobs is not the CEO of Apple Computer, Inc.

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Problem 71

Technology For Exercises $69-74$ , let $a$ represent the statement "Apple Computer, Inc. developed the iPad," and $j$ represent the statement, "Steve Jobs is the CEO of Apple Computer, Inc." Convert each compound statement into symbols.

Apple Computer, Inc, did not develop the iPad, or Steve Jobs is the CEO of Apple Computer, Inc.

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Problem 72

Technology For Exercises $69-74$ , let $a$ represent the statement "Apple Computer, Inc. developed the iPad," and $j$ represent the statement, "Steve Jobs is the CEO of Apple Computer, Inc." Convert each compound statement into symbols.

Apple Computer, Inc. developed the iPad, or Steve Jobs is the CEO of Apple Computer, Inc.

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Problem 73

Technology For Exercises $69-74$ , let $a$ represent the statement "Apple Computer, Inc. developed the iPad," and $j$ represent the statement, "Steve Jobs is the CEO of Apple Computer, Inc." Convert each compound statement into symbols.

Suppose the statements that Apple Computer, Inc. developed the iPad and that Steve Jobs is CEO of Apple Computer, Inc. are both true. Which of Exercises $69-72$ are true statements?

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Problem 74

Suppose the statements that Apple Computer, Inc. developed the iPad and that Steve Jobs is CEO of Apple Computer, Inc. are both false. Which of Exercises $69-72$ are true statements?

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Problem 75

Medicine The following excerpts appear in a home medical reference book. Source: American College of Physicians Complete Home Medical Guide.
Which of these excerpts are statements?

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Problem 76

Medicine The following excerpts appear in a home medical reference book. Source: American College of Physicians Complete Home Medical Guide.
Which of these excerpts are compound statements?

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Problem 77

Medicine The following excerpts appear in a home medical reference book. Source: American College of Physicians Complete Home Medical Guide.
Write the negation of excerpt e.

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Problem 78

Law The following excerpts appear in a guide to common laws. Source: Law for Dummies.
Which of these excerpts are statements?

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Problem 79

Law The following excerpts appear in a guide to common laws. Source: Law for Dummies.
Which of these excepts are compound statements?

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Problem 80

Law The following excerpts appear in a guide to common laws. Source: Law for Dummies.
Write the negation of excerpt e.

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Problem 81

Philosophy Read each of the following quotes from ancient philosophers. Provide an argument why these quotes may or may not be called statements. Source: Masterworks of Philosophy.
a. "A friend is a friend of someone."-Socrates
b. "Every art, and every science reduced to a teachable form, and in like manner every action and moral choice, aims, it is thought, at some good: for which reason a common and by no means a bad description of what the Chief Good is, 'that which all things aim at." "- Aristotle
c. “Furthermore, Friendship helps the young to keep from error: the old, in respect of attention and such deficiencies in action as their weakness makes them liable to; and those who are in their prime, in respect of noble deeds, because they are thus more able to devise plans and carry them out.”—Aristotle

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Problem 82

Bible Read each of the following quotes from the biblical book Proverbs. Provide an argument why these quotes may or may not be called statements. Source: TNIV Bible.
a. "A gentle answer turns away wrath." -Proverbs $15 : 1$
b. "The hot-tempered stir up dissension, but those who are patient calm a quarrel." Proverbs $15 : 18$
c. "When justice is done, it brings joy to the righteous but terror to evildoers," - Proverbs $21 : 15$
d. "Do not exploit the poor because they are poor and do not crush the needy in court."- Proverbs $22 : 22$
e. "Apply your heart to instruction and your ears to words of knowledge."- Proverbs $23 : 12$

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Problem 83

Football For Exercises $83-88,$ let $n$ represent the statement "New Orleans won the Super Bowl" and $m$ represent the statement "Peyton Manning is the best quarterback." Convert each compound statement into symbols.

New Orleans won the Super Bowl but Peyton Manning is not the best quarterback.

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Problem 84

Football For Exercises $83-88,$ let $n$ represent the statement "New Orleans won the Super Bowl" and $m$ represent the statement "Peyton Manning is the best quarterback." Convert each compound statement into symbols.

New Orleans did not win the Super Bowl or Peyton Manning is not the best quarterback.

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Problem 85

Football For Exercises $83-88,$ let $n$ represent the statement "New Orleans won the Super Bowl" and $m$ represent the statement "Peyton Manning is the best quarterback." Convert each compound statement into symbols.

New Orleans did not win the Super Bowl or Peyton Manning is not the best ouarterback.

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Problem 86

Football For Exercises $83-88,$ let $n$ represent the statement "New Orleans won the Super Bowl" and $m$ represent the statement "Peyton Manning is the best quarterback." Convert each compound statement into symbols.

New Orleans did not win the Super Bowl but Peyton Manning is the best quarterback.

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Problem 87

Football For Exercises $83-88,$ let $n$ represent the statement "New Orleans won the Super Bowl" and $m$ represent the statement "Peyton Manning is the best quarterback." Convert each compound statement into symbols.

Neither did New Orleans win the Super Bowl nor is Peyton Manning the best quarterback.

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Problem 88

Football For Exercises $83-88,$ let $n$ represent the statement "New Orleans won the Super Bowl" and $m$ represent the statement "Peyton Manning is the best quarterback." Convert each compound statement into symbols.

Either New Orleans won the Super Bowl or Peyton Manning is the best quarterback, and it is not the case that both New Orleans won the Super Bowl and Peyton Manning is the best quarterback.

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Problem 89

APPLY IT Suppose the statements that New Orleans won the Super Bowl and that Peyton Manning is the best quarterback are both true. Which of Exercises $83-88$ are true statements?

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Problem 90

Suppose the statements that New Orleans won the Super Bowl and that Peyton Manning is the best quarterback are both false. Which of Exercises $83-88$ are true statements?

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