Multiple choice

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?

  1. I and III

  2. I and IV

  3. II and III

  4. II and IV
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using logical equivalences: ¬∀x(P(x)) ≡ ∃x(¬P(x)) (De Morgan for quantifiers). So I and IV are equivalent. II and III are also equivalent to each other (¬∃x(P(x)) ≡ ∀x(¬P(x))), but they're not equivalent to I and IV.

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