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

Hence (i) and (iv) are equal.

Ans (D)

Answer 2

B is correct for this I and IV are equivalent.

Answer 3

.