Chapter 1

The Foundations: Logic and Proofs

Educators

FY

Problem 1

Which of these sentences are propositions? What are the truth values of those that are propositions?
a) Boston is the capital of Massachusetts.
b) Miami is the capital of Florida.
c) $2+3=5$
d) $5+7=10$
e) $x+2=11$
f) Answer this question.

James C.
Numerade Educator

Problem 2

Which of these are propositions? What are the truth values of those that are propositions?
a) Do not pass go.
b) What time is it?
c) There are no black flies in Maine.
d) $4+x=5$ .
e) The moon is made of green cheese.
f) $2^{n} \geq 100$ .

FY
Fan Y.
Numerade Educator

Problem 3

What is the negation of each of these propositions?
a) Linda is younger than Sanjay.
b) Mei makes more money than Isabella.
c) Moshe is taller than Monica.
d) Abby is richer than Ricardo.

James C.
Numerade Educator

Problem 4

What is the negation of each of these propositions?
a) Janice has more Facebook friends than Juan.
b) Quincy is smarter than Venkat.
c) Zelda drives more miles to school than Paola.
d) Briana sleeps longer than Gloria.

FY
Fan Y.
Numerade Educator

Problem 5

What is the negation of each of these propositions?
a) Mei has an MP3 player.
b) There is no pollution in New Jersey.
c) $2+1=3$ .
d) The summer in Maine is hot and sunny.

James C.
Numerade Educator

Problem 6

What is the negation of each of these propositions?
a) Jennifer and Teja are friends.
b) There are 13 items in a baker’s dozen.
c) Abby sent more than 100 text messages yesterday.
d) 121 is a perfect square.

FY
Fan Y.
Numerade Educator

Problem 7

What is the negation of each of these propositions?
a) Steve has more than 100 $\mathrm{GB}$ free disk space on his laptop.
b) Zach blocks e-mails and texts from Jennifer.
c) $7 \cdot 11 \cdot 13=999$ .
d) Diane rode her bicycle 100 miles on Sunday.

James C.
Numerade Educator

Problem 8

Suppose that Smartphone A has 256 MB RAM and 32 GB ROM, and the resolution of its camera is 8 MP; Smartphone B has 288 MB RAM and 64 GB ROM, and the resolution of its camera is 4 MP; and Smartphone C has 128 MB RAM and 32 GB ROM, and the resolution of its camera is 5 MP. Determine the truth value of each of these propositions.
a) Smartphone B has the most RAM of these three smartphones.
b) Smartphone C has more ROM or a higher resolution camera than Smartphone B.
c) Smartphone B has more RAM, more ROM, and a higher resolution camera than Smartphone A.
d) If Smartphone B has more RAM and more ROM than Smartphone C, then it also has a higher resolution camera.
e) Smartphone A has more RAM than Smartphone B if and only if Smartphone B has more RAM than Smartphone A.

FY
Fan Y.
Numerade Educator

Problem 9

Suppose that during the most recent fiscal year, the annual revenue of Acme Computer was 138 billion dollars and its net profit was 8 billion dollars, the annual revenue of Nadir Software was 87 billion dollars and its net profit was 5 billion dollars, and the annual revenue of Quixote Media was 111 billion dollars and its net profit was 13 billion dollars. Determine the truth value of each of these propositions for the most recent fiscal year.
a) Quixote Media had the largest annual revenue.
b) Nadir Software had the lowest net profit and Acme Computer had the largest annual revenue.
c) Acme Computer had the largest net profit or Quixote Media had the largest net profit.
d) If Quixote Media had the smallest net profit, then Acme Computer had the largest annual revenue.
e) Nadir Software had the smallest net profit if and only if Acme Computer had the largest annual revenue.

James C.
Numerade Educator

Problem 10

Let $p$ and $q$ be the propositions
$p :$ I bought a lottery ticket this week.
$q :$ I won the million dollar jackpot.
Express each of these propositions as an English sentence.
a) $\neg p$
b) $p \vee q$
c) $p \rightarrow q$
d) $p \wedge q$
e) $p \leftrightarrow q$
f) $\neg p \rightarrow \neg q$
g) $\neg p \wedge \neg q$
h) $\neg p \vee(p \wedge q)$

FY
Fan Y.
Numerade Educator

Problem 11

Let $p$ and $q$ be the propositions "Swimming at the New Jersey shore is allowed" and "Sharks have been spotted near the shore," respectively. Express each of these compound propositions as an English sentence.
a) $\neg q$
b) $p \vee q$
c) $\neg p \vee q$
d) $p \rightarrow \neg q$
e) $\neg q \rightarrow p$
f) $\neg p \rightarrow \neg q$
g) $p \leftrightarrow \neg q$
h) $\neg p \wedge(p \vee \neg q)$

James C.
Numerade Educator

Problem 12

Let $p$ and $q$ be the propositions "The election is decided" and "The votes have been counted, " respectively. Express each of these compound propositions an English sentence.
a) $\neg p$
b) $p \vee q$
c) $\neg p \wedge q$
d) $q \rightarrow p$
e) $\neg q \rightarrow \neg p$
f) $\neg p \rightarrow \neg q$
g) $p \leftrightarrow q$
h) $\neg q \vee(\neg p \wedge q)$

FY
Fan Y.
Numerade Educator

Problem 13

Let $p$ and $q$ be the propositions
$p :$ It is below freezing.
$q :$ It is snowing.
Write these propositions using $p$ and $q$ and logical connectives (including negations).
a) It is below freezing and snowing.
b) It is below freezing but not snowing.
c) It is not below freezing and it is not snowing.
d) It is either snowing or below freezing (or both).
e) If it is below freezing, it is also snowing.
f ) Either it is below freezing or it is snowing, but it is not snowing if it is below freezing.
g) That it is below freezing is necessary and sufficient for it to be snowing.

James C.
Numerade Educator

Problem 14

Let $p, q,$ and $r$ be the propositions
$p :$ You have the flu.
$q :$ You miss the final examination.
$r :$ You pass the course.
Express each of these propositions as an English sentence.
a) $p \rightarrow q$
b) $\neg q \leftrightarrow r$
c) $q \rightarrow \neg r$
d) $p \vee q \vee r$
e) $(p \rightarrow \neg r) \vee(q \rightarrow \neg r)$
f) $(p \wedge q) \vee(\neg q \wedge r)$

FY
Fan Y.
Numerade Educator

Problem 15

Let $p$ and $q$ be the propositions
$p :$ You drive over 65 miles per hour.
$q :$ You get a speeding ticket.
Write these propositions using $p$ and $q$ and logical connectives (including negations).
a) You do not drive over 65 miles per hour.
b) You drive over 65 miles per hour, but you do not get a speeding ticket.
c) You will get a speeding ticket if you drive over 65 miles per hour.
d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket.
e) Driving over 65 miles per hour is sufficient for getting a speeding ticket.
f ) You get a speeding ticket, but you do not drive over 65 miles per hour.
g) Whenever you get a speeding ticket, you are driving over 65 miles per hour.

James C.
Numerade Educator

Problem 16

Let $p, q,$ and $r$ be the propositions
$p :$ You get an $\mathrm{A}$ on the final exam.
$q :$ You do every exercise in this book.
$r :$ You get an $\mathrm{A}$ in this class.
Write these propositions using $p, q,$ and $r$ and logical connectives (including negations).
a) You get an A in this class, but you do not do every exercise in this book.
b) You get an A on the final, you do every exercise in this book, and you get an A in this class.
c) To get an A in this class, it is necessary for you to get an A on the final.
d) You get an A on the final, but you don’t do every exercise in this book; nevertheless, you get an A in this class.
e) Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class.
f) You will get an A in this class if and only if you either do every exercise in this book or you get an A on the final.

FY
Fan Y.
Numerade Educator

Problem 17

Let $p, q,$ and $r$ be the propositions
$p :$ Grizzly bears have been seen in the area.
$q :$ Hiking is safe on the trail.
$r :$ Berries are ripe along the trail.
a) Berries are ripe along the trail, but grizzly bears have not been seen in the area.
b) Grizzly bears have not been seen in the area and hiking on the trail is safe, but berries are ripe along the trail.
c) If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area.
d) It is not safe to hike on the trail, but grizzly bears have not been seen in the area and the berries along the trail are ripe.
e) For hiking on the trail to be safe, it is necessary but not sufficient that berries not be ripe along the trail and for grizzly bears not to have been seen in the area.
f) Hiking is not safe on the trail whenever grizzly bears have been seen in the area and berries are ripe along the trail.

James C.
Numerade Educator

Problem 18

Determine whether these biconditionals are true or false.
a) $2+2=4$ if and only if $1+1=2$
b) $1+1=2$ if and only if $2+3=4$
c) $1+1=3$ if and only if monkeys can fly.
d) $0>1$ if and only if $2>1$

FY
Fan Y.
Numerade Educator

Problem 19

Determine whether each of these conditional statements is true or false.
a) If $1+1=2,$ then $2+2=5$
b) If $1+1=3,$ then $2+2=4$
c) If $1+1=3,$ then $2+2=5$
d) If monkeys can $\mathrm{fly},$ then $1+1=3$

James C.
Numerade Educator

Problem 20

Determine whether each of these conditional statements is true or false.
a) If $1+1=3,$ then unicorns exist.
b) If $1+1=3,$ then dogs can fly.
c) If $1+1=2,$ then dogs can fly.
d) If $2+2=4,$ then $1+2=3$

FY
Fan Y.
Numerade Educator

Problem 21

For each of these sentences, determine whether an inclusive or, or an exclusive or, is intended. Explain your answer.
a) Coffee or tea comes with dinner.
b) A password must have at least three digits or be at least eight characters long.
c) The prerequisite for the course is a course in number theory or a course in cryptography.
d) You can pay using U.S. dollars or euros.

James C.
Numerade Educator

Problem 22

For each of these sentences, determine whether an inclusive or, or an exclusive or, is intended. Explain your answer.
a) Experience with $\mathrm{C}++$ or Java is required.
b) Lunch includes soup or salad.
c) To enter the country you need a passport or a voter registration card.
d) Publish or perish.

FY
Fan Y.
Numerade Educator

Problem 23

For each of these sentences, state what the sentence means if the logical connective or is an inclusive or (that is, a disjunction) versus an exclusive or. Which of these meanings of or do you think is intended?
a) To take discrete mathematics, you must have taken calculus or a course in computer science.
b) When you buy a new car from Acme Motor Company, you get $\$ 2000$ back in cash or a 2$\%$ car loan.
c) Dinner for two includes two items from column A or three items from column B.
d) School is closed if more than two feet of snow falls or if the wind chill is below $-100^{\circ} \mathrm{F} .$

James C.
Numerade Educator

Problem 24

Write each of these statements in the form "if $p,$ then $q^{\prime \prime}$ in English. [Hint: Refer to the list of common ways to express conditional statements provided in this section.]
a) It is necessary to wash the boss's car to get promoted.
b) Winds from the south imply a spring thaw.
c) A sufficient condition for the warranty to be good is that you bought the computer less than a year ago.
d) Willy gets caught whenever he cheats.
e) You can access the website only if you pay a subscription fee.
f) Getting elected follows from knowing the right people.
g) Carol gets seasick whenever she is on a boat.

FY
Fan Y.
Numerade Educator

Problem 25

Write each of these statements in the form “if p, then q” in English. [Hint: Refer to the list of common ways to express conditional statements.]
a) It snows whenever the wind blows from the northeast.
b) The apple trees will bloom if it stays warm for a week.
c) That the Pistons win the championship implies that they beat the Lakers.
d) It is necessary to walk eight miles to get to the top of Long’s Peak.
e) To get tenure as a professor, it is sufficient to be world famous.
f) If you drive more than 400 miles, you will need to buy gasoline.
g) Your guarantee is good only if you bought your CD player less than 90 days ago.
h) Jan will go swimming unless the water is too cold.
i) We will have a future, provided that people believe in science.

James C.
Numerade Educator

Problem 26

Write each of these statements in the form “if p, then q” in English. [Hint: Refer to the list of common ways to express conditional statements provided in this section.]
a) I will remember to send you the address only if you send me an e-mail message.
b) To be a citizen of this country, it is sufficient that you were born in the United States.
c) If you keep your textbook, it will be a useful reference in your future courses.
d) The Red Wings will win the Stanley Cup if their goalie plays well.
e) That you get the job implies that you had the best credentials.
f) The beach erodes whenever there is a storm.
g) It is necessary to have a valid password to log on to the server.
h) You will reach the summit unless you begin your climb too late.
i) You will get a free ice cream cone, provided that you are among the first 100 customers tomorrow.

FY
Fan Y.
Numerade Educator

Problem 27

Write each of these propositions in the form “p if and only if q” in English.
a) If it is hot outside you buy an ice cream cone, and if you buy an ice cream cone it is hot outside.
b) For you to win the contest it is necessary and sufficient that you have the only winning ticket.
c) You get promoted only if you have connections, and you have connections only if you get promoted.
d) If you watch television your mind will decay, and conversely.
e) The trains run late on exactly those days when I take it.

James C.
Numerade Educator

Problem 28

Write each of these propositions in the form “p if and only if q” in English.
a) For you to get an A in this course, it is necessary and sufficient that you learn how to solve discrete mathematics problems.
b) If you read the newspaper every day, you will be informed, and conversely.
c) It rains if it is a weekend day, and it is a weekend day if it rains.
d) You can see the wizard only if the wizard is not in, and the wizard is not in only if you can see him.
e) My airplane flight is late exactly when I have to catch a connecting flight.

FY
Fan Y.
Numerade Educator

Problem 29

State the converse, contrapositive, and inverse of each of these conditional statements.
a) If it snows today, I will ski tomorrow.
b) I come to class whenever there is going to be a quiz.
c) A positive integer is a prime only if it has no divisors other than 1 and itself.

James C.
Numerade Educator

Problem 30

State the converse, contrapositive, and inverse of each of these conditional statements.
a) If it snows tonight, then I will stay at home.
b) I go to the beach whenever it is a sunny summer day.
c) When I stay up late, it is necessary that I sleep until noon.

FY
Fan Y.
Numerade Educator

Problem 31

How many rows appear in a truth table for each of these compound propositions?
a) $p \rightarrow \neg p$
b) $(p \vee \neg r) \wedge(q \vee \neg s)$
c) $q \vee p \vee \neg s \vee \neg r \vee \neg t \vee u$
d) $(p \wedge r \wedge t) \leftrightarrow(q \wedge t)$

James C.
Numerade Educator

Problem 32

How many rows appear in a truth table for each of these compound propositions?
a) $(q \rightarrow \neg p) \vee(\neg p \rightarrow \neg q)$
b) $(p \vee \neg t) \wedge(p \vee \neg s)$
c) $(p \rightarrow r) \vee(\neg s \rightarrow \neg t) \vee(\neg u \rightarrow v)$
d) $(p \wedge r \wedge s) \vee(q \wedge t) \vee(r \wedge \neg t)$

FY
Fan Y.
Numerade Educator

Problem 33

Construct a truth table for each of these compound propositions.
a) $p \wedge \neg p$
b) $p \vee \neg p$
c) $(p \vee \neg q) \rightarrow q$
d) $(p \vee q) \rightarrow(p \wedge q)$
e) $(p \rightarrow q) \leftrightarrow(\neg q \rightarrow \neg p)$
f) $(p \rightarrow q) \rightarrow(q \rightarrow p)$

James C.
Numerade Educator

Problem 34

Construct a truth table for each of these compound propositions.
a. $p \rightarrow \neg p$
b) $p \leftrightarrow \neg p$
c. $p \oplus(p \vee q)$
d) $(p \wedge q) \rightarrow(p \vee q)$
e. $(q \rightarrow \neg p) \leftrightarrow(p \leftrightarrow q)$
f. $(p \leftrightarrow q) \oplus(p \leftrightarrow \neg q)$

FY
Fan Y.
Numerade Educator

Problem 35

Construct a truth table for each of these compound propositions.
a) $p \rightarrow \neg p$
b) $(p \oplus q) \rightarrow(p \wedge q)$
c) $p \oplus(p \vee q)$
d) $(p \leftrightarrow q) \oplus(\neg p \leftrightarrow q)$
e) $(p \leftrightarrow q) \oplus(\neg p \leftrightarrow \neg r)$
f) $(p \oplus q) \rightarrow(p \oplus \neg q)$

Bernabe M.
Numerade Educator

Problem 36

Construct a truth table for each of these compound propositions.
a) $p \oplus p$
b) $p \oplus \neg p$
c) $p \oplus \neg q$
d) $\neg p \oplus \neg q$
e) $(p \oplus q) \vee(p \oplus \neg q)$
f) $(p \oplus q) \wedge(p \oplus \neg q)$

FY
Fan Y.
Numerade Educator

Problem 37

Construct a truth table for each of these compound propositions.
a. $p \rightarrow \neg q$
b) $\neg p \leftrightarrow q$
c. $(p \rightarrow q) \vee(\neg p \rightarrow q)$
d) $(p \rightarrow q) \wedge(\neg p \rightarrow q)$
e. $(p \leftrightarrow q) \vee(\neg p \rightarrow q)$
f. $(\neg p \leftrightarrow \neg q) \leftrightarrow(p \leftrightarrow q)$

Bernabe M.
Numerade Educator

Problem 38

Construct a truth table for each of these compound propositions.
a. $(p \vee q) \vee r$
b) $(p \vee q) \wedge r$
c) $(p \wedge q) \vee r$
d) $(p \wedge q) \wedge r$
e) $(p \vee q) \wedge \neg r$
f) $(p \wedge q) \vee \neg r$

FY
Fan Y.
Numerade Educator

Problem 39

Construct a truth table for each of these compound propositions.
a) $p \rightarrow(\neg q \vee r)$
b) $\neg p \rightarrow(q \rightarrow r)$
c) $(p \rightarrow q) \vee(\neg p \rightarrow r)$
d) $(p \rightarrow q) \wedge(\neg p \rightarrow r)$
e) $(p \rightarrow q) \vee(\neg q \rightarrow r)$
f) $(\neg p \leftrightarrow \neg q) \leftrightarrow(q \leftrightarrow r)$

Bernabe M.
Numerade Educator

Problem 40

Construct a truth table for $((p \rightarrow q) \rightarrow r) \rightarrow s$

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Fan Y.
Numerade Educator

Problem 41

Construct a truth table for $(p \leftrightarrow q) \leftrightarrow(r \leftrightarrow s)$

James C.
Numerade Educator

Problem 42

Explain, without using a truth table, why $(p \vee \neg q) \wedge$ $(q \vee \neg r) \wedge(r \vee \neg p)$ is true when $p, q,$ and $r$ have the same truth value and it is false otherwise.

FY
Fan Y.
Numerade Educator

Problem 43

Explain, without using a truth table, why $(p \vee q \vee r) \wedge$ $(\neg p \vee \neg q \vee \neg r)$ is true when at least one of $p, q,$ and $r$ is true and at least one is false, but is false when all three variables have the same truth value.

James C.
Numerade Educator

Problem 44

If $p_{1}, p_{2}, \ldots, p_{n}$ are $n$ propositions, explain why
$$
\bigwedge_{i=1}^{n-1} \bigwedge_{j=i+1}^{n}\left(\neg p_{i} \vee \neg p_{j}\right)
$$
is true if and only if at most one of $p_{1}, p_{2}, \ldots, p_{n}$ is true.

Check back soon!

Problem 45

Use Exercise 44 to construct a compound proposition that is true if and only if exactly one of the propositions $p_{1}, p_{2}, \ldots, p_{n}$ is true. [Hint: Combine the compound proposition in Exercise 44 and a compound proposition that is true if and only if at least one of $p_{1}, p_{2}, \ldots, p_{n}$ is true.]

James C.
Numerade Educator

Problem 46

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Fan Y.
Numerade Educator

Problem 47

James C.
Numerade Educator

Problem 48

Evaluate each of these expressions.
a) 1000$\wedge(01011 \vee 11011)$
b) $(01111 \wedge 10101) \vee 01000$
c) $(01010 \oplus 11011) \oplus 01000$
d) $(11011 \vee 01010) \wedge(10001 \vee 1011)$

Bernabe M.
Numerade Educator

Problem 49

Fuzzy logic is used in artificial intelligence. In fuzzy logic, a proposition has a truth value that is a number between 0 and 1, inclusive. A proposition with a truth value of 0 is false and one with a truth value of 1 is true. Truth values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value 0.8 can be assigned to the statement “Fred is happy,” because Fred is happy most of the time, and the truth value 0.4 can be assigned to the statement “John is happy,” because John is happy slightly less than half the time. Use these truth values to solve

The truth value of the negation of a proposition in fuzzy logic is 1 minus the truth value of the proposition. What are the truth values of the statements “Fred is not happy” and “John is not happy”?

James C.
Numerade Educator

Problem 50

Fuzzy logic is used in artificial intelligence. In fuzzy logic, a proposition has a truth value that is a number between 0 and 1, inclusive. A proposition with a truth value of 0 is false and one with a truth value of 1 is true. Truth values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value 0.8 can be assigned to the statement “Fred is happy,” because Fred is happy most of the time, and the truth value 0.4 can be assigned to the statement “John is happy,” because John is happy slightly less than half the time. Use these truth values to solve

The truth value of the conjunction of two propositions in fuzzy logic is the minimum of the truth values of the two propositions. What are the truth values of the statements “Fred and John are happy” and “Neither Fred nor John is happy”?

FY
Fan Y.
Numerade Educator

Problem 51

Fuzzy logic is used in artificial intelligence. In fuzzy logic, a proposition has a truth value that is a number between 0 and 1, inclusive. A proposition with a truth value of 0 is false and one with a truth value of 1 is true. Truth values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value 0.8 can be assigned to the statement “Fred is happy,” because Fred is happy most of the time, and the truth value 0.4 can be assigned to the statement “John is happy,” because John is happy slightly less than half the time. Use these truth values to solve

The truth value of the disjunction of two propositions in fuzzy logic is the maximum of the truth values of the two propositions. What are the truth values of the statements “Fred is happy, or John is happy” and “Fred is not happy, or John is not happy”?

James C.
Numerade Educator

Problem 52

Fuzzy logic is used in artificial intelligence. In fuzzy logic, a proposition has a truth value that is a number between 0 and 1, inclusive. A proposition with a truth value of 0 is false and one with a truth value of 1 is true. Truth values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value 0.8 can be assigned to the statement “Fred is happy,” because Fred is happy most of the time, and the truth value 0.4 can be assigned to the statement “John is happy,” because John is happy slightly less than half the time. Use these truth values to solve

Is the assertion “This statement is false” a proposition?

FY
Fan Y.
Numerade Educator

Problem 53

Fuzzy logic is used in artificial intelligence. In fuzzy logic, a proposition has a truth value that is a number between 0 and 1, inclusive. A proposition with a truth value of 0 is false and one with a truth value of 1 is true. Truth values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value 0.8 can be assigned to the statement “Fred is happy,” because Fred is happy most of the time, and the truth value 0.4 can be assigned to the statement “John is happy,” because John is happy slightly less than half the time. Use these truth values to solve

The $n$ th statement in a list of 100 statements is "Exactly $n$ of the statements in this list are false."
a) What conclusions can you draw from these state- ments?
b) Answer part (a) if the $n$ th statement is "At least $n$ of the statements in this list are false.'
c) Answer part (b) assuming that the list contains 99 statements.

Bernabe M.
Numerade Educator

Problem 54

Fuzzy logic is used in artificial intelligence. In fuzzy logic, a proposition has a truth value that is a number between 0 and 1, inclusive. A proposition with a truth value of 0 is false and one with a truth value of 1 is true. Truth values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value 0.8 can be assigned to the statement “Fred is happy,” because Fred is happy most of the time, and the truth value 0.4 can be assigned to the statement “John is happy,” because John is happy slightly less than half the time. Use these truth values to solve

An ancient Sicilian legend says that the barber in a remote town who can be reached only by traveling a dangerous mountain road shaves those people, and only those people, who do not shave themselves. Can there be such a barber?

Bernabe M.
Numerade Educator