Let P, Q and R be three atomic prepositional assertions. Let X denote (P v Q)…
2005
Let P, Q and R be three atomic prepositional assertions. Let X denote (P v Q) → R and Y denote (P → R) v (Q → R). Which one of the following is a tautology?
- A.
X ≡ Y
- B.
X → Y
- C.
Y → X
- D.
¬ Y → X
Attempted by 84 students.
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Correct answer: B
Answer: X → Y is a tautology.
Reason:
Rewrite X: (P ∨ Q) → R ≡ ¬(P ∨ Q) ∨ R = (¬P ∧ ¬Q) ∨ R = (¬P ∨ R) ∧ (¬Q ∨ R) = (P → R) ∧ (Q → R).
Rewrite Y: (P → R) ∨ (Q → R).
Thus X → Y becomes ((P → R) ∧ (Q → R)) → ((P → R) ∨ (Q → R)), which is always true because whenever both conjuncts hold at least one disjunct holds.
Why the other choices fail (counterexamples):
Equivalence of X and Y fails: let P = true, Q = false, R = false. Then X is false but Y is true, so X ≡ Y is not a tautology.
Y → X fails with the same assignment (P = true, Q = false, R = false): Y is true while X is false, so the implication is false.
¬Y → X fails: let P = true, Q = true, R = false. Then Y is false (so ¬Y is true) while X is false, making the implication false.