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Mathematics-Online course: Basic Mathematics - Propositional Logic

Quantifiers

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The phrases
,,there exists ...``,    and     ,,for all ...``
are symbolically abbreviated by the existential quantifier $ \exists$ and the universal quantifier $ \forall$ respectively. These quantifiers are commonly used in the context of statements $ A(p)$ that depend on a parameter $ p$ in a set $ P$ .

Notation meaning
$ \exists\,p\in P:\ A(p)$ there is at least one $ p$ in $ P$, for which $ A(p)$ is true
$ \forall\,p \in P:\ A(p)$ $ A(p)$ is true for all $ p$ in $ P$

Negation of statements turns existential quantifiers into universial quantifiers and vice versa:

$\displaystyle \lnot\big( \exists\,p\in P:\ A(p) \big)$ $\displaystyle =$ $\displaystyle \forall\, p\in P:\ \lnot A(p)$  
$\displaystyle \lnot\big( \forall\,p\in P:\ A(p) \big)$ $\displaystyle =$ $\displaystyle \exists\, p\in P:\ \lnot A(p)$  

The symbol $ \exists!$ is also commonly used to represent the phrase ,,there exists one and only one ...``.

(Authors: Höllig/Kimmerle/Abele)

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  automatically generated 10/31/2008