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If I have $\forall x: P(x)$, what is the equivalent function if I attach a not to the whole function:

$\neg (\forall x: P(x))$

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up vote 3 down vote accepted

Read the statement out: not (for all x, p(x) holds). So if not for all x, there must be some x such that p(x) doesn't hold. So:

Then you have $\exists x : \lnot P(x)$.

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So the existential qualifier is the inverse of the universal quantifier? –  David Feb 5 at 6:56
    
No, read what I wrote carefully. If that were true, it wouldn't make any sense. –  Newb Feb 5 at 6:57
    
Oh I understand what you mean. So it would be like me saying not(for all x, x=2), therefore there exists some x, not(x=2) –  David Feb 5 at 6:59
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you'll have $$\exists x: \neg P(x)$$

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