Suppose the predicate F(x, y, t) is used to represent the statement that person x can fool person y at time t. Which one of the statements below expresses best the meaning of the formula, ~(forall x)(exists y)(forall t) F(x, y, t) A. Everyone can fool some person at some time B. No one can fool everyone all the time C. Everyone cannot fool some person all the time D. No one can fool some person at some time

GATE 2010 · Discrete Mathematics · First Order Logic · easy

Answer: B. No one can fool everyone all the time

1. Apply quantifier negation laws: ~(forall x)(exists y)(forall t) F(x,y,t) = (exists x) ~(exists y)(forall t) F(x,y,t) = (exists x)(forall y) ~(forall t) F(x,y,t) = (exists x)(forall y)(exists t) ~F(x,y,t)
2. Read the resulting formula in English: There exists a person x such that for every person y there is some time t when x cannot fool y. In other words, there is some person who cannot fool everyone at all times — equivalently, no one can fool everyone all the time.