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+17 votes

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
in Mathematical Logic by Boss (17.5k points)
edited by | 1.2k views
Remember negation of quantifiers-

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

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

4 Answers

+15 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

$I \implies$ 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.

by Boss (16.3k points)
edited by
+12 votes
I and IV are equal
by Boss (14.4k points)
+5 votes
Do double negation of (i) which gives (iv).

Hence Option B is Ans.
by Boss (23.9k points)
edited by
+4 votes

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 $(B).$

by Veteran (58.6k points)
edited by

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