C S 2813 Fall 2025
Homework #1
(Due on 9/19 @ 11:59 PM)
Q.1) [8 pt] Which of these are propositions?
a) Do not pass go.
b) What time is it?
c) 4 + x = 5.
d) The moon is made of green cheese.
Q.2) [8 pt] 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) p ∨ q
b) p → q
c) p ↔ q.
d) ¬p ∧ (p ∨ ¬q).
Q.3) [6 pt] Translate the given statement into propositional logic using the
propositions provided. To use the wireless network in the airport you must pay
the daily fee unless you are a subscriber to the service. Express your answer in
terms of w: “You can use the wireless network in the airport,” d: “You pay the
daily fee,” and s: “You are a subscriber to the service.”
Q.4) [6 pt] Are these system specifications consistent? “If the file system is
not locked, then new messages will be queued. If the file system is not locked,
then the system is functioning normally, and conversely. If new messages are
not queued, then they will be sent to the message buffer. If the file system is
not locked, then new messages will be sent to the message buffer. New messages
will not be sent to the message buffer.”
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Q.5) [6 pt] Use truth tables to verify the associative laws.
a) (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
b) (p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
Q.6) [10 pt] Show that each of the following conditional statements is a
tautology by applying a chain of logical identities. (Do not use truth tables).
a) [p ∧ (p → q)] → q
b) [(p ∨ q) ∧ (p → r) ∧ (q → r)] → r
Q.7) [8 pt] Let N (x) be the statement “x has visited North Dakota,” where
the domain consists of the students in your school. Express each of these quantifications in English.
a) ∃xN (x)
b) ∀xN (x)
c) ¬∀xN (x)
d) ∀x¬N (x)
Q.8) [8 pt] Determine the truth value of each of these statements if the
domain of each variable consists of all real numbers.
a) ∃x(x2 = 2)
b) ∃x(x2 = −1)
c) ∀x(x2 + 2 ≥ 1)
d) ∀x(x2 ̸= x)
Q.9) [8 pt] Let P (x, y) be the statement “Student x has taken class y,” where
the domain for x consists of all students in your class and for y consists of all
computer science courses at your school. Express each of these quantifications
in English.
a) ∃x∃yP (x, y)
b) ∃x∀yP (x, y)
c) ∃y∀xP (x, y)
d) ∀x∀yP (x, y)
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Q.10) [4 pt] Express the negations of each of these statements so that all
negation symbols immediately precede predicates.
a) ∃x∃yP (x, y) ∧ ∀x∀yQ(x, y)
b) ∃x∃y(Q(x, y) ↔ Q(y, x))
Q.11) [10 pt] For each of these arguments use rule of inference to show the
hypotheses imply the conclusion.
a) “Linda, a student in this class, owns a red convertible. Everyone who
owns a red convertible has gotten at least one speeding ticket. Therefore,
someone in this class has gotten a speeding ticket.”
b) “There is someone in this class who has been to France. Everyone who
goes to France visits the Louvre. Therefore, someone in this class has
visited the Louvre.”
Q.12) [6 pt] Use a direct proof to show that the product of two odd numbers
is odd.
Q.13) [6 pt] Prove or disprove that the product of a nonzero rational number
and an irrational number is irrational.
Q.14) [6 pt] Use a proof by cases to show that 10 is not the square of a
positive integer. [Hint: Consider two cases: (i) 1 ≤ x ≤ 3, (ii) x ≥ 4.]
References:
• Kenneth Rosen, Discrete Mathematics and Its Applications, McGraw Hill,
2025. Website
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