Negation of for every and there exist statements [duplicate]

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I need to find the negations of the following statements:

For every n (is an element of natural numbers) there is a prime number p such that n < p.

There is n (is an element of natural numbers) such that for every prime number p, p < n.

What are the rules of negation for these statements? Thanks!

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1 Answer

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The negation of $\forall x.P$ is $\exists x .(\neg P)$.

Similarly, we have $\neg (\exists x .P) \equiv \forall x (.\neg P)$.

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