CSC 014: Exercise 1 - Spring 2024
Answers.
(1) (a) Not a proposition.
(b) Proposition. False?
(c) Not a proposition.
(d) (Compound) Proposition. True.
(e) Proposition. False
(f) Proposition. True.
(2) (a) I didn’t buy a lottery ticket this week.
(b) I bought a lottery ticket this week or I won the million-dollar jackpot.
(c) I bought a lottery ticket this week and I won the million-dollar jackpot.
(d) I won the million-dollar jackpot if I bought a lottery ticket this week.
(e) I didn’t buy a lottery ticket this week and I didn’t win the million-dollar jackpot.
(3) (a) r ∧ ¬q
(b) p ∧ q ∧ r.
(c) q → p
(d) p → q
(4) r ↔ (p ∨ q)
(5) Evaluate the following compound propositions, given that p = T, q = F, r = T, s = F .
(a) (p ∧ ¬q) = (T ∧ ¬F ) = (T ∧ T ) = T
(b) (r ∧ s) → (p ∨ q) = (T ∧ F ) → (T ∨ F ) = F → T = T
(c) (p → (¬s))∧(s → q)∧((¬q) → p) = (T → (¬F ))∧(F → F )∧((¬F ) → T ) = (T → T )∧T ∧(T →
T ) = T ∧ T ∧ T.
(6) Use truth tables to prove that:
(a) p → q and (¬p) ∨ q are equivalent.
p q p → q ¬p (¬p) ∨ q
T T
T
F
T
T F
F
F
F
F T
T
T
T
F F
T
T
T
Since the columns for p → q and (¬p) ∨ q are identical, the propositions are equivalent.
(b) ¬(p ∧ ¬q) and ¬p ∨ q are equivalent.
p q ¬q p ∧ ¬q ¬(p ∧ ¬q) ¬p ¬p ∨ q
T T F
F
T
F
T
T F T
T
F
F
F
F T F
F
T
T
T
F F T
F
T
T
T
Since the columns for ¬(p ∧ ¬q) and ¬p ∨ q are identical, the propositions are equivalent.
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