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Previous Year Question (PYQs)

Let P denote “She is intelligent” and Q denote “She is happy.”
Given statements:
(a) If she is intelligent, then she is unhappy.
(b) She is neither intelligent nor happy.
(c) It is necessary to be not intelligent in order to be happy.
(d) To be not intelligent is to be unhappy.

Which option gives the correct propositional expressions?

  1. (a) P→¬Q; (b) P ∧ ¬Q; (c) Q→¬P; (d) P→¬Q
  2. (a) P→Q; (b) ¬P ∧ ¬Q; (c) ¬Q→P; (d) ¬P→¬Q
  3. (a) P→¬Q; (b) ¬P ∧ ¬Q; (c) Q→¬P; (d) ¬P→¬Q
  4. (a) P→¬Q; (b) P ∧ ¬Q; (c) Q↔P; (d) ¬P→¬Q




Solution

Translate each statement:
(a) “If intelligent then unhappy” ⇒ P → ¬Q.
(b) “Neither intelligent nor happy” ⇒ ¬P ∧ ¬Q.
(c) “Being not intelligent is necessary for being happy” ⇒ Q → ¬P.
(d) “Not intelligent implies unhappy” ⇒ ¬P → ¬Q.

Only Option 3 matches all four translations.

Answer: 3


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