Negations of quantifiers satisfy the following rules:

\(\neg \forall x \in S, P(x) = \exists x \in S \ni \neg P(x)\)

\(\neg \exists x \in S \ni P(x) = \forall x \in S, \neg P(x)\)

In particular, if the negation passes through a quantifier, the quantifier should flip.

Example of negating \(\forall\)

Let \(S\) be the set \(S = \{0, 3, 5\}\) and let \(P(x)\) be the statement \(P(x) = \)"\(x < 4.\)" Then, the following are both true because they are logically equivalent:

\(\neg \forall x \in S, P(x) =\) "Not all elements of \(x\) are less than \(4.\)"

\(\exists x \in S \ni \neg P(x) =\) "There exists an element of \(S\) that is not less than \(4.\)"

Both statements are true because \(5 \in S\) and \(5 \not< 4.\)

Example of negating \(\exists\)

Let \(S\) be the set of all people and let \(P(x)\) be the statement \(P(x) = \)"\(x\) eats \(100\) pounds of pizza every day." Then, the following are both true because they are logically equivalent:

\(\neg \exists x \in S \ni P(x) =\) "There does not exist any person who eats \(100\) pounds of pizza every day."

\(\forall x \in S, \neg P(x) =\) "Every person does not eat \(100\) pounds of pizza every day."

Both statements are true because nobody eats \(100\) pounds of pizza daily.

Which is equivalent to \(\neg \forall x \in U, Q(x)?\)

Which is equivalent to \(\exists x \in V, \neg S(x)?\)