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Consider the following well-formed formulae:

1. $\neg \forall x(P(x))$
2. $\neg \exists x(P(x))$
3. $\neg \exists x(\neg P(x))$
4. $\exists x(\neg P(x))$

Which of the above are equivalent?

1. I and III
2. I and IV
3. II and III
4. II and IV
edited | 826 views

Option (B) is correct.  I and IV are equivalent.

$¬∀x(P(x)) \equiv ∃x(¬P(x))$    [De morgan's Law]

Alternate approach:

Let's take an example.

Let $P(x)\implies$  Student $x$ is pass

$I \implies$ Not all students are pass. (which means "Some students are fail")

$II\implies$There does not exist a student who is pass. (which means "Every student is fail")

$III \implies$There does not exist a student who is not pass  (which means "Every student is pass")

$IV\implies$Some students are not pass. (which means "Some students are fail")

I and IV are equivalent.

answered by Boss (13.3k points)
edited by
I and IV are equal
answered by Boss (14.3k points)
Do double negation of (i) which gives (iv).

Hence Option B is Ans.
answered by Boss (22.9k points)
edited

Using De morgan's Law

answered by Loyal (9.6k points)
+1 vote

Just take negation out in iv then option i and iv is same so option b is correct

answered by Loyal (6.8k points)