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.
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.
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.
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.
Let and 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) ¬q
b) p ∧ q
c) ¬p ∨ q
d) p → ¬q
e) ¬q → p
f) ¬p → ¬q
g) p ↔ ¬q
h) ¬p ∧ (p∨ ¬q)
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.
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.
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.
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 newcar fromAcmeMotor 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 .
How many rows appear in a truth table for each of these
compound propositions?
a) p → ¬p
b) (p ∨ ¬r) ∧ (q ∨ ¬s)
c) q ∨ p ∨ ¬s ∨ ¬r ∨ ¬t ∨ u
d) (p ∧ r ∧ t) ↔ (q ∧ t)
a) p ∧ ¬p
b) p ∨ ¬p
c) (p ∨ ¬q) → q
d) (p ∨ q) → (p ∧ q)
e) (p → q) ↔ (¬q → ¬p)
f) (p → q) → (q → p)
Construct a truth table for each of these compound propositions.
a) (p ∨ q) → (p ⊕ q)
b) (p ⊕ q) → (p ∧ q)
c) (p ∨ q) ⊕ (p ∧ q)
d) (p ↔ q) ⊕ (¬p ↔ q)
e) (p ↔ q) ⊕ (¬p ↔ ¬r)
f) (p ⊕ q) → (p ⊕ ¬q)
Construct a truth table for (p ↔ q) ↔ (r ↔ s).