GATE CSE 2009 | Question: 26

Question

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?

• A. $\text{I}$ and $\text{III}$
• B. $\text{I}$ and $\text{IV}$
• C. $\text{II}$ and $\text{III}$
• D. $\text{II}$ and $\text{IV}$

Comments

Remember negation of quantifiers-

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

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

Answer 1

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.

Answer 2

I and IV are equal

Answer 3

Do double negation of (i) which gives (iv).

Hence Option B is Ans.

Answer 4

Using De Morgan's Law

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