Definition:Existential Quantifier/Unique
Definition
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.
Definition 1
There exists a unique object $x$ such that $\map P x$, denoted $\exists ! x: \map P x$, if and only if:
- $\exists x : \paren {\map P x \land \forall y : \paren {\map P y \implies x = y} }$
In natural language, this means:
- There exists exactly one $x$ with the property $P$
- is logically equivalent to:
Definition 2
There exists a unique object $x$ such that $\map P x$, denoted $\exists ! x: \map P x$, if and only if:
- $\exists x : \forall y : \paren {\map P y \iff x = y}$
Definition 3
There exists a unique object $x$ such that $\map P x$, denoted $\exists ! x: \map P x$, if and only if both:
- $\exists x : \map P x$
and:
- $\forall y : \forall z : \paren {\paren {\map P y \land \map P z} \implies y = z }$
The symbol $\exists !$ is a variant of the existential quantifier $\exists$: there exists at least one.
Also denoted as
The symbol $\exists_1$ is also found for the same concept, being an instance of the exact existential quantifier $\exists_n$.
Some sources, for example 1972: Patrick Suppes: Axiomatic Set Theory, use $\operatorname E !$, which is idiosyncratic, considering the use in the same source of $\exists$ for the general existential quantifier.
Also known as
Some sources refer to this as the uniqueness quantifier.
Also see
- Results about the unique existential quantifier can be found here.
Sources
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Variables and quantifiers
- 1972: Patrick Suppes: Axiomatic Set Theory (2nd ed.) ... (previous) ... (next): $\S 1.2$ Logic and Notation