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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 | 668 views

Option B is correct.  I and IV are equivalent.

¬∀x(P(x)) = ∃x(¬P(x))    [De morgan's Law]

Alternate approach:

Let's take an example.

Let P(x) = Student x is pass.

I→ Not all students are pass. (which means "Some students are fail")

II→There doesn't exist a student who is pass. (which means "Every student is fail")

III→There doesn't exist a student who is not pass  (which means "Every student is pass")

IV→Some students are not pass. (which means "Some students are fail")

I and IV are equivalent.

selected by
I and IV are equal
Do double negation of (i) which gives (iv).

Hence Option B is Ans.
edited
+1 vote

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

+1 vote

Using De morgan's Law