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Existential quantification
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== Basics == Consider the [[formal logic|formal]] sentence :For some natural number <math>n</math>, <math>n\times n=25</math>. This is a single statement using existential quantification. It is roughly analogous to the informal sentence "Either <math>0\times 0=25</math>, or <math>1\times 1=25</math>, or <math>2\times 2=25</math>, or... and so on," but more precise, because it doesn't need us to infer the meaning of the phrase "and so on." (In particular, the sentence explicitly specifies its [[domain of discourse]] to be the natural numbers, not, for example, the [[real number]]s.) This particular example is true, because 5 is a natural number, and when we substitute 5 for ''n'', we produce the true statement <math>5\times 5=25</math>. It does not matter that "<math>n\times n=25</math>" is true only for that single natural number, 5; the existence of a single [[solution (equation)|solution]] is enough to prove this existential quantification to be true. In contrast, "For some [[even number]] <math>n</math>, <math>n\times n=25</math>" is false, because there are no even solutions. The [[domain of discourse]], which specifies the values the variable ''n'' is allowed to take, is therefore critical to a statement's trueness or falseness. [[Logical conjunction]]s are used to restrict the domain of discourse to fulfill a given predicate. For example, the sentence :For some positive odd number <math>n</math>, <math>n\times n=25</math> is [[logically equivalent]] to the sentence :For some natural number <math>n</math>, <math>n</math> is odd and <math>n\times n=25</math>. The [[mathematical proof]] of an existential statement about "some" object may be achieved either by a [[constructive proof]], which exhibits an object satisfying the "some" statement, or by a [[nonconstructive proof]], which shows that there must be such an object without concretely exhibiting one. ===Notation=== In [[First-order logic|symbolic logic]], "β" (a turned letter "[[E]]" in a [[sans-serif]] font, Unicode U+2203) is used to indicate existential quantification. For example, the notation <math>\exists{n}{\in}\mathbb{N}: n\times n=25</math> represents the (true) statement :There exists some <math>n</math> in the set of [[natural number]]s such that <math>n\times n=25</math>. The symbol's first usage is thought to be by [[Giuseppe Peano]] in ''[[Formulario mathematico]]'' (1896). Afterwards, [[Bertrand Russell]] popularised its use as the existential quantifier. Through his research in set theory, Peano also introduced the symbols <math>\cap</math> and <math>\cup</math> to respectively denote the intersection and union of sets.<ref name="Webb2018">{{cite book |author=Stephen Webb |title=Clash of Symbols |publisher=Springer Cham |year=2018 |isbn=978-3-319-71349-6 |doi=10.1007/978-3-319-71350-2 |url=http://link.springer.com/10.1007/978-3-319-71350-2|pages=210β211}}</ref>
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