Remember negation of quantifiers-

$\neg\forall x(P(x))=\exists x(\neg P(x))$

$\neg \exists x(P(x))=\forall x(\neg P(x))$

$\neg\forall x(P(x))=\exists x(\neg P(x))$

$\neg \exists x(P(x))=\forall x(\neg P(x))$

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22 votes

Consider the following well-formed formulae:

- $\neg \forall x(P(x))$
- $\neg \exists x(P(x))$
- $\neg \exists x(\neg P(x))$
- $\exists x(\neg P(x))$

Which of the above are equivalent?

- $\text{I}$ and $\text{III}$
- $\text{I}$ and $\text{IV}$
- $\text{II}$ and $\text{III}$
- $\text{II}$ and $\text{IV}$

20 votes

Best answer

**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

- $\text{I}\;\; \implies$ Not all students are pass. (which means "Some students are fail")
- $\text{II}\;\implies$There does not exist a student who is pass. (which means "Every student is fail")
- $\text{III} \implies$There does not exist a student who is not pass (which means "Every student is pass")
- $\text{IV} \implies$Some students are not pass. (which means "Some students are fail")

I and IV are equivalent.

4 votes

Using De Morgan's Law

- $\neg \forall x\big(P(x)\big) \equiv \exists x\big(\neg P(x) \big) $
- $ \neg \exists x\big( P(x) \big) \equiv \forall x\big(\neg P(x) \big)$
- $ \neg \exists x\big( \neg P(x) \big) \equiv \forall x\big(P(x) \big)$
- $ \exists x\big( \neg P(x) \big) \equiv \neg \forall x\big(P(x) \big)$

$I$ and $IV$ are equivalents.

So, the correct answer is $(B).$