Page 34 - Discrete Mathematics and Its Applications

1.1 Propositional Logic 13

5. What is the negation of each of these propositions? 10. Let p and q be the propositions “The election is decided”
a) Steve has more than 100 GB free disk space on his and “The votes have been counted,” respectively. Express
laptop. each of these compound propositions as an English sen-
b) Zach blocks e-mails and texts from Jennifer. tence.
c) 7 · 11 · 13 = 999. a) ¬p b) p ∨ q
d) Diane rode her bicycle 100 miles on Sunday. c) ¬p ∧ q d) q → p
6. Suppose that SmartphoneA has 256 MB RAM and 32 GB e) ¬q →¬p f) ¬p →¬q
ROM, and the resolution of its camera is 8 MP; Smart- g) p ↔ q h) ¬q ∨ (¬p ∧ q)
phone B has 288 MB RAM and 64 GB ROM, and the 11. Let p and q be the propositions
resolution of its camera is 4 MP; and Smartphone C has p : It is below freezing.
128 MB RAM and 32 GB ROM, and the resolution of q : It is snowing.
its camera is 5 MP. Determine the truth value of each of
Write these propositions using p and q and logical con-
these propositions.
nectives (including negations).
a) Smartphone B has the most RAM of these three smart-
a) It is below freezing and snowing.
phones. b) It is below freezing but not snowing.
b) Smartphone C has more ROM or a higher resolution c) It is not below freezing and it is not snowing.
camera than Smartphone B.
c) Smartphone B has more RAM, more ROM, and a d) It is either snowing or below freezing (or both).
e) If it is below freezing, it is also snowing.
higher resolution camera than Smartphone A.
d) If Smartphone B has more RAM and more ROM than f) Either it is below freezing or it is snowing, but it is
Smartphone C, then it also has a higher resolution not snowing if it is below freezing.
camera. g) That it is below freezing is necessary and sufficient
e) Smartphone A has more RAM than Smartphone B if for it to be snowing.
and only if Smartphone B has more RAM than Smart- 12. Let p, q, and r be the propositions
phone A. p : You have the flu.
7. Suppose that during the most recent fiscal year, the an- q : You miss the final examination.
nual revenue of Acme Computer was 138 billion dollars r : You pass the course.
and its net profit was 8 billion dollars, the annual revenue Express each of these propositions as an English sen-
of Nadir Software was 87 billion dollars and its net profit tence.
was 5 billion dollars, and the annual revenue of Quixote a) p → q b) ¬q ↔ r
Media was 111 billion dollars and its net profit was c) q →¬r d) p ∨ q ∨ r
13 billion dollars. Determine the truth value of each of e) (p →¬r) ∨ (q →¬r)
these propositions for the most recent fiscal year. f) (p ∧ q) ∨ (¬q ∧ r)
a) Quixote Media had the largest annual revenue. 13. Let p and q be the propositions
b) Nadir Software had the lowest net profit and Acme
p : You drive over 65 miles per hour.
Computer had the largest annual revenue. q : You get a speeding ticket.
c) Acme Computer had the largest net profit or Quixote
Media had the largest net profit. Write these propositions using p and q and logical con-
d) If Quixote Media had the smallest net profit, then nectives (including negations).
Acme Computer had the largest annual revenue. a) You do not drive over 65 miles per hour.
e) Nadir Software had the smallest net profit if and only b) You drive over 65 miles per hour, but you do not get
if Acme Computer had the largest annual revenue. a speeding ticket.
c) You will get a speeding ticket if you drive over
8. Let p and q be the propositions
65 miles per hour.
p : I bought a lottery ticket this week. d) If you do not drive over 65 miles per hour, then you
q : I won the million dollar jackpot.
will not get a speeding ticket.
Express each of these propositions as an English sen- e) Driving over 65 miles per hour is sufficient for getting
tence. a speeding ticket.
a) ¬p b) p ∨ q c) p → q f) You get a speeding ticket, but you do not drive over
d) p ∧ q e) p ↔ q f) ¬p →¬q 65 miles per hour.
g) ¬p ∧¬q h) ¬p ∨ (p ∧ q) g) Whenever you get a speeding ticket, you are driving
9. Let p and q be the propositions “Swimming at the New over 65 miles per hour.
Jersey shore is allowed” and “Sharks have been spotted 14. Let p, q, and r be the propositions
near the shore,” respectively. Express each of these com- p : You get an A on the final exam.
pound propositions as an English sentence. q : You do every exercise in this book.
a) ¬q b) p ∧ q c) ¬p ∨ q r : You get an A in this class.
d) p →¬q e) ¬q → p f) ¬p →¬q Write these propositions using p, q, and r and logical
g) p ↔¬q h) ¬p ∧ (p ∨¬q)
connectives (including negations).