Exercise 1152Logic, sets and proofs › Quantifiers and negationL1Added 3 Nov 2026

De Morgan's laws for quantifiers

#De Morgan#quantifiers#negation

Statement

State and prove the two De Morgan laws for quantifiers:
$\neg(\forall x,\ P(x))\equiv\exists x,\ \neg P(x),\qquad \neg(\exists x,\ P(x))\equiv\forall x,\ \neg P(x).$

Hint

For the first: if not all satisfy $P$ , some fails $P$ . For the second: if no $x$ satisfies $P$ , all fail $P$ .

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Solution

First law. $\neg(\forall x,\ P(x))$ means "it is not the case that every $x$ satisfies $P$ ", i.e. some $x$ does not satisfy $P$ : $\exists x,\ \neg P(x)$ .

Second law. $\neg(\exists x,\ P(x))$ means "no $x$ satisfies $P$ ", i.e. every $x$ fails $P$ : $\forall x,\ \neg P(x)$ .

Conclusion. Both laws hold; they generalize De Morgan's laws for connectives.

De Morgan's laws for quantifiers

Statement

Solution

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