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Return to GATE2009 q26.
Consider the following well-formed formulae:
I. $\neg \forall x(P(x))$
II. $\neg \exists x(P(x))$
III. $\neg \exists x(\neg P(x))$
IV. $\exists x(\neg P(x))$
Which of the above are equivalent?
(A) I and III
(B) I and IV
(C) II and III
(D) II and IV
A formula $∀x(P(x))$ is equivalent to formula $¬∃x(¬P(x))$ i.e. add $¬$ inside and outside, and convert $∀$ to $∃$.
So, $¬∀x(P(x))$ is equivalent to $∃x(¬P(x))$.
GATECSE