Page 35 - Discrete Mathematics and Its Applications

14 1 / The Foundations: Logic and Proofs

a) You get an A in this class, but you do not do every a) Coffee or tea comes with dinner.
exercise in this book. b) A password must have at least three digits or be at
b) You get anA on the final, you do every exercise in this least eight characters long.
book, and you get an A in this class. c) The prerequisite for the course is a course in number
c) To get an A in this class, it is necessary for you to get theory or a course in cryptography.
an A on the final. d) You can pay using U.S. dollars or euros.
d) You get an A on the final, but you don’t do every ex- 20. For each of these sentences, determine whether an in-
ercise in this book; nevertheless, you get an A in this
clusive or, or an exclusive or, is intended. Explain your
class.
answer.
e) Getting an A on the final and doing every exercise in
a) Experience with C++ or Java is required.
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 b) Lunch includes soup or salad.
do every exercise in this book or you get an A on the c) To enter the country you need a passport or a voter
final. registration card.
15. Let p, q, and r be the propositions d) Publish or perish.
p : Grizzly bears have been seen in the area. 21. Foreachofthesesentences,statewhatthesentencemeans
q : Hiking is safe on the trail. if the logical connective or is an inclusive or (that is, a dis-
r : Berries are ripe along the trail. junction) versus an exclusive or.Which of these meanings
of or do you think is intended?
Write these propositions using p, q, and r and logical
connectives (including negations). a) To take discrete mathematics, you must have taken
calculus or a course in computer science.
a) Berries are ripe along the trail, but grizzly bears have
not been seen in the area. b) When you buy a new car fromAcme Motor Company,
b) Grizzly bears have not been seen in the area and hik- you get $2000 back in cash or a 2% car loan.
ing on the trail is safe, but berries are ripe along the c) Dinner for two includes two items from column A or
trail. three items from column B.
c) If berries are ripe along the trail, hiking is safe if and d) School is closed if more than 2 feet of snow falls or if
only if grizzly bears have not been seen in the area. the wind chill is below −100.
d) It is not safe to hike on the trail, but grizzly bears have 22. Write each of these statements in the form “if p, then q”
not been seen in the area and the berries along the trail in English. [Hint: Refer to the list of common ways to ex-
are ripe. press conditional statements provided in this section.]
e) For hiking on the trail to be safe, it is necessary but not a) It is necessary to wash the boss’s car to get promoted.
sufficient that berries not be ripe along the trail and b) Winds from the south imply a spring thaw.
for grizzly bears not to have been seen in the area. c) A sufficient condition for the warranty to be good is
f) Hiking is not safe on the trail whenever grizzly bears that you bought the computer less than a year ago.
have been seen in the area and berries are ripe along
the trail. d) Willy gets caught whenever he cheats.
e) You can access the website only if you pay a subscrip-
16. Determine whether these biconditionals are true or tion fee.
false.
f) Getting elected follows from knowing the right peo-
a) 2 + 2 = 4 if and only if 1 + 1 = 2.
ple.
b) 1 + 1 = 2 if and only if 2 + 3 = 4.
g) Carol gets seasick whenever she is on a boat.
c) 1 + 1 = 3 if and only if monkeys can fly.
d) 0 > 1 if and only if 2 > 1. 23. Write each of these statements in the form “if p, then q”
in English. [Hint: Refer to the list of common ways to
17. Determine whether each of these conditional statements express conditional statements.]
is true or false.
a) It snows whenever the wind blows from the northeast.
a) If 1 + 1 = 2, then 2 + 2 = 5.
b) If 1 + 1 = 3, then 2 + 2 = 4. b) The apple trees will bloom if it stays warm for a week.
c) If 1 + 1 = 3, then 2 + 2 = 5. c) That the Pistons win the championship implies that
d) If monkeys can fly, then 1 + 1 = 3. they beat the Lakers.
d) It is necessary to walk 8 miles to get to the top of
18. Determine whether each of these conditional statements
Long’s Peak.
is true or false.
e) To get tenure as a professor, it is sufficient to be world-
a) If 1 + 1 = 3, then unicorns exist.
famous.
b) If 1 + 1 = 3, then dogs can fly.
c) If 1 + 1 = 2, then dogs can fly. f) If you drive more than 400 miles, you will need to buy
d) If 2 + 2 = 4, then 1 + 2 = 3. gasoline.
g) Your guarantee is good only if you bought your CD
19. For each of these sentences, determine whether an in-
clusive or, or an exclusive or, is intended. Explain your player less than 90 days ago.
answer. h) Jan will go swimming unless the water is too cold.