Definition:Existential Quantifier/Exact

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Definition

The symbol $\exists_n$ denotes the existence of an exact number of objects fulfilling a particular condition.

$\exists_n x: \map P x$

means:

There exist exactly $n$ objects $x$ such that $\map P x$ holds.


It is a variant of the existential quantifier $\exists$: there exists at least one.


Unique Quantifier

The symbol $\exists !$ denotes the existence of a unique object fulfilling a particular condition.

$\exists ! x: \map P x$

means:

There exists exactly one object $x$ such that $\map P x$ holds

or:

There exists one and only one $x$ such that $\map P x$ holds.


This quantifier is called the unique existential quantifier.