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

GATE 2013 · Discrete Mathematics · First Order Logic · easy

Answer: D. ~exists x (F(x) ^ P(x))

1. Rewrite 'none' as 'there is no': 'None of my friends are perfect' = 'There is no x that is both my friend and perfect' = ~exists x (F(x) ^ P(x)).
2. Verify equivalence with option A: Option A, forall x (F(x) -> ~P(x)), is logically equivalent to option D. However, option D is the most direct translation of the sentence structure 'none are', and the GATE key marks D as the answer. Both A and D are logically equivalent, but D more literally captures 'none'.