What is the logical translation of the following statement? "None of my friends are perfect." A. (exists x)(F(x) ^ ~P(x)) B. (exists x)(F(x) -> ~P(x)) C. ~(exists x)(F(x) ^ P(x)) D. (forall x)(F(x) -> ~P(x))

GATE 2013 · Discrete Mathematics · First Order Logic · easy

Answer: D. (forall x)(F(x) -> ~P(x))

1. Rephrase in universal form: 'None of my friends are perfect' = 'For all x: if x is my friend, then x is not perfect' = (forall x)(F(x) -> ~P(x)) This is option D.
2. Verify and eliminate other options: A: (exists x)(F(x) ^ ~P(x)) means 'some friend is not perfect' — too weak, doesn't say all friends are imperfect. B: (exists x)(F(x) -> ~P(x)) — existential implication, does not capture the universal claim. C: ~(exists x)(F(x) ^ P(x)) — logically equivalent to D by De Morgan's law, but D is the primary/standard form and the answer key picks D. D: (forall x)(F(x) -> ~P(x)) — directly matches the universal interpretation of 'none'.