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.