Uniqueness

Uniqueness is a property ascribed to a particular object in a set concerning a property possessed by said object, but not by any other. For example, zero is the unique additive identity in the real numbers because it is precisely the only real number that serves as the additive identity.

Uniqueness is often used in conjunction with the existential quantifier in predicate logic, and symbolized by $\exists!$. Given a set $S$ with a property $P$, we express the existence and uniqueness of an element $x$ satisfying $P$ by:

$\exists!x\in S: P\left(x\right)$ ("There exists a unique $x$ in $S$ such that $P\left(x\right)$")

To reduce the language to simple existential and universal quantification, we recognize that there is indeed an element $x$ satisfying $P$ (itself an existential statement) and that for any element $y$, if $y$ also satisfies $P$, then $y$ must be the same as $x$:

$\exists x\in S:\left(P\left(x\right)\And\forall y:\left(P\left(y\right)\implies x=y\right)\right)$.

An equivalent definition that is more commonly used in proof separates the notions of existence and uniqueness together, first by expressing the existence of $x$ satisfying $S$ and then expressing that any two elements satisfying $S$ must actually be the same:

$\exists x\in S:P\left(x\right)\And\forall y,z\in S:\left(P\left(y\right)\And P\left(z\right)\implies y=z\right)$.