Editing Universal quantification (section)

=== Negation ===
The negation of a universally quantified function is obtained by changing the universal quantifier into an [[existential quantifier]] and negating the quantified formula. That is, 
:<math>\lnot \forall x\; P(x)\quad\text {is equivalent to}\quad \exists x\;\lnot P(x) </math>
where <math>\lnot</math> denotes [[negation]].

For example, if {{math|''P''(''x'')}} is the [[propositional function]] "{{math|''x''}} is married", then, for the [[set (mathematics)|set]] {{mvar|X}} of all living human beings, the universal quantification
<blockquote>Given any living person {{math|''x''}}, that person is married</blockquote>
is written
:<math>\forall x \in X\, P(x)</math>

This statement is false. Truthfully, it is stated that
<blockquote>It is not the case that, given any living person {{mvar|''x''}}, that person is married</blockquote>
or, symbolically:
:<math>\lnot\ \forall x \in X\, P(x)</math>.

If the function {{math|''P''(''x'')}} is not true for ''every'' element of {{mvar|X}}, then there must be at least one element for which the statement is false. That is, the negation of <math>\forall x \in X\, P(x)</math> is logically equivalent to "There exists a living person {{math|''x''}} who is not married", or:
:<math>\exists x \in X\, \lnot P(x)</math>

It is erroneous to confuse "all persons are not married" (i.e. "there exists no person who is married") with "not all persons are married" (i.e. "there exists a person who is not married"):
:<math>\lnot\ \exists x \in X\, P(x) \equiv\ \forall x \in X\, \lnot P(x) \not\equiv\ \lnot\ \forall x\in X\, P(x) \equiv\ \exists x \in X\, \lnot P(x)</math>