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Discrete Mathematics and its Applications

Kenneth H. Rosen

Chapter 1

The Foundations: Logic and Proofs - all with Video Answers

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Section 1

Propositional Logic

03:42

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.

Willis James
Willis James
Numerade Educator
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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) $2^{n} \geq 100$

Alice Simper
Alice Simper
Numerade Educator
01:01

Problem 3

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 Chok
James Chok
Numerade Educator
04:37

Problem 4

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 every day.
d) 121 is a perfect square.

Samuel Goyette
Samuel Goyette
Numerade Educator
01:01

Problem 5

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 Chok
James Chok
Numerade Educator
05:10

Problem 6

Suppose that Smartphone A has $256 \mathrm{MB}$ RAM and 32 GB ROM, and the resolution of its camera is $8 \mathrm{MP}$; Smartphone $\mathrm{B}$ has $288 \mathrm{MB}$ RAM and 64 GB ROM, and the resolution of its camera is $4 \mathrm{MP}$; and Smartphone $\mathrm{C}$ 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 $\mathrm{B}$ has the most RAM of these three smartphones.
b) Smartphone $\mathrm{C}$ has more ROM or a higher resolution camera than Smartphone $\mathrm{B}$.
c) Smartphone $\mathrm{B}$ 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 $\mathrm{C}$, then it also has a higher resolution camera.
e) Smartphone A has more RAM than Smartphone $\mathrm{B}$ if and only if Smartphone B has more RAM than Smartphone $\mathrm{A}$.

Fan Yang
Fan Yang
Numerade Educator
01:25

Problem 7

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 Chok
James Chok
Numerade Educator
08:09

Problem 8

Let $p$ and $q$ be the propositions $p:$ I bought a lottery ticket this week. $q: \mathrm{I}$ 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) $\neg p \vee(p \wedge q)$

Fan Yang
Fan Yang
Numerade Educator
08:09

Problem 9

Let $p$ and $q$ be the propositions $p:$ I bought a lottery ticket this week. $q: \mathrm{I}$ 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) $\neg p \vee(p \wedge q)$

Fan Yang
Fan Yang
Numerade Educator
04:09

Problem 10

Let $p$ and $q$ be the propositions 'The election is decided" and "The votes have been counted," respectively. Express each of these compound propositions as 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) $\neg q \vee(\neg p \wedge q)$

Clayton Schubring
Clayton Schubring
Numerade Educator
01:55

Problem 11

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 Chok
James Chok
Numerade Educator
05:36

Problem 12

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) $(p \wedge q) \vee(\neg q \wedge r)$

Fan Yang
Fan Yang
Numerade Educator
01:57

Problem 13

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 Chok
James Chok
Numerade Educator
01:24

Problem 14

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 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 $\mathrm{A}$ on the final, you do every exercise in this book, and you get an A in this class.
c) To get an $\mathrm{A}$ in this class, it is necessary for you to get an $\mathrm{A}$ on the final.
d) You get an $\mathrm{A}$ on the final, but you don't do every exercise in this book; nevertheless, you get an $\mathrm{A}$ in this class.
e) Getting an $\mathrm{A}$ 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 $\mathrm{A}$ on the final.

Jennifer Stoner
Jennifer Stoner
Numerade Educator
01:48

Problem 15

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. Write these propositions using $p, q$, and $r$ and logical connectives (including negations).
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.

Jennifer Stoner
Jennifer Stoner
Numerade Educator
03:04

Problem 16

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 $2>1$.

Fan Yang
Fan Yang
Numerade Educator
01:20

Problem 17

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 fly, then $1+1=3$.

James Chok
James Chok
Numerade Educator
01:11

Problem 19

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 Chok
James Chok
Numerade Educator
01:19

Problem 20

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.

Fan Yang
Fan Yang
Numerade Educator
04:20

Problem 21

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 $\mathrm{B}$.
d) School is closed if more than 2 feet of snow falls or if the wind chill is below $-100$.

Jacob Kogan
Jacob Kogan
Numerade Educator
02:55

Problem 22

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) 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.

Jennifer Stoner
Jennifer Stoner
Numerade Educator
04:55

Problem 23

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 8 miles to get to the top of Long's Peak.
e) To get tenure as a professor, it is sufficient to be worldfamous.
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.f

Jennifer Stoner
Jennifer Stoner
Numerade Educator
04:40

Problem 23

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 8 miles to get to the top of Long's Peak.
e) To get tenure as a professor, it is sufficient to be worldfamous.
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.

Jennifer Stoner
Jennifer Stoner
Numerade Educator
04:39

Problem 24

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

Jennifer Stoner
Jennifer Stoner
Numerade Educator
01:45

Problem 25

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 Chok
James Chok
Numerade Educator
02:25

Problem 26

Write each of these propositions in the form " $p$ if and only if $q$ " in English.
a) For you to get an $\mathrm{A}$ 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.

Jennifer Stoner
Jennifer Stoner
Numerade Educator
02:31

Problem 27

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 Chok
James Chok
Numerade Educator
04:59

Problem 28

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.

Fan Yang
Fan Yang
Numerade Educator
01:08

Problem 29

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

Manisha Sarker
Manisha Sarker
Numerade Educator
01:14

Problem 30

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) $(p \wedge r \wedge s) \vee(q \wedge t) \vee(r \wedge \neg t)$

Fan Yang
Fan Yang
Numerade Educator
20:36

Problem 31

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) $(p \rightarrow q) \rightarrow(q \rightarrow p)$

Bernabe Montoya
Bernabe Montoya
Numerade Educator
17:09

Problem 32

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) $(p \leftrightarrow q) \oplus(p \leftrightarrow \neg q)$

Willis James
Willis James
Numerade Educator
19:50

Problem 33

Construct a truth table for each of these compound propositions.
a) $(p \vee q) \rightarrow(p \oplus q)$
b) $(p \oplus q) \rightarrow(p \wedge q)$
c) $(p \vee q) \oplus(p \wedge q)$
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 Montoya
Bernabe Montoya
Numerade Educator
04:49

Problem 34

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) $(p \oplus q) \wedge(p \oplus \neg q)$

WM
William Mead
Numerade Educator
20:36

Problem 35

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) $(p \leftrightarrow q) \vee(\neg p \leftrightarrow q)$
f) $(\neg p \leftrightarrow \neg q) \leftrightarrow(p \leftrightarrow q)$

Bernabe Montoya
Bernabe Montoya
Numerade Educator
17:07

Problem 36

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) $(p \wedge q) \vee \neg r$

Willis James
Willis James
Numerade Educator
20:36

Problem 37

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) $(p \leftrightarrow q) \vee(\neg q \leftrightarrow r)$
f) $(\neg p \leftrightarrow \neg q) \leftrightarrow(q \leftrightarrow r)$

Bernabe Montoya
Bernabe Montoya
Numerade Educator
02:06

Problem 38

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

James Chok
James Chok
Numerade Educator
04:20

Problem 39

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

Sanat Mukherjee
Sanat Mukherjee
Numerade Educator
02:59

Problem 40

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.

Fan Yang
Fan Yang
Numerade Educator
01:58

Problem 41

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 Chok
James Chok
Numerade Educator
01:18

Problem 42

What is the value of $x$ after each of these statements is encountered in a computer program, if $x=1$ before the statement is reached?
a) if $x+2=3$ then $x:=x+1$
b) if $(x+1=3)$ OR $(2 x+2=3)$ then $x:=x+1$
c) if $(2 x+3=5)$ AND $(3 x+4=7)$ then $x:=x+1$
d) if $(x+1=2)$ XOR $(x+2=3)$ then $x:=x+1$
e) if $x<2$ then $x:=x+1$

Jennifer Stoner
Jennifer Stoner
Numerade Educator
03:15

Problem 43

Find the bitwise $O R$, bitwise $A N D$, and bitwise $X O R$ of each of these pairs of bit strings.
a) 1011110,0100001
b) 11110000,10101010
$\begin{array}{lllllll}\text { c) } 00 & 0111 & 000 & 1 & 10 & 0100 & 1000\end{array}$
d) 1111111111,0000000000

Jennifer Stoner
Jennifer Stoner
Numerade Educator
01:47

Problem 44

Evaluate each of these expressions.
a) $11000 \wedge(01011 \vee 11011)$
b) $(01111 \wedge 10101) \vee 01000$
c) $(01010 \oplus 11011) \oplus 01000$
d) $(11011 \vee 01010) \wedge(10001 \vee 11011)$

Jennifer Stoner
Jennifer Stoner
Numerade Educator
01:18

Problem 45

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 Exercises $45-47$.
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?"

Manisha Sarker
Manisha Sarker
Numerade Educator
01:42

Problem 46

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 Exercises $45-47$.
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?"

Manisha Sarker
Manisha Sarker
Numerade Educator
01:34

Problem 47

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 Exercises $45-47$.
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?"

Manisha Sarker
Manisha Sarker
Numerade Educator
01:19

Problem 48

Is the assertion "This statement is false" a proposition?

Nick Johnson
Nick Johnson
Numerade Educator
00:33

Problem 49

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 statements?
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.

Yujie Wang
Yujie Wang
College of San Mateo
01:00

Problem 50

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?

Nick Johnson
Nick Johnson
Numerade Educator