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GATE 2010 CS (CS1) – Question 40

Engineering Mathematics · Discrete Mathematics: Propositional and First Order Logic · 2 marks · Multiple choice

Let F(x,y,t) mean that x can fool y at time t. What does $\forall x\,\exists y\,\exists t\,\neg F(x,y,t)$ express?

  1. Everyone can fool some person at some time
  2. No one can fool everyone all the time
  3. Everyone cannot fool some person all the time
  4. No one can fool some person at some time

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Correct answer: (B) No one can fool everyone all the time

Explanation

For every person x there is some person y and some time t at which x fails to fool y. Equivalently, negating the whole statement yields $\exists x\forall y\forall t\,F(x,y,t)$: someone fools everyone all the time. The original is its negation, B.

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