Exercise 1862Logic, sets and proofs › Quantifiers and negationL1Added 3 Nov 2026

Negating a complex statement

#negation#practice#nested

Statement

Negate: $\forall n\in\mathbb{N},\ \exists p>n,\ p$ is prime.

Hint

$\neg\forall\to\exists$ , $\neg\exists\to\forall$ , negate the predicate.

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Negation. $\exists n\in\mathbb{N},\ \forall p>n,\ p$ is not prime.

Interpretation. "There is some $n$ such that every integer larger than $n$ is composite." (False, by infinitely many primes.)

Conclusion. The negation is as above.