# Dictionary Definition

uniqueness n : the quality of being one of a
kind; "that singularity distinguished him from all his companions"
[syn: singularity]

# User Contributed Dictionary

#### Translations

state or quality of being unique or
one-of-a-kind

- Czech: jedinečnost

# Extensive Definition

In mathematics and logic, the phrase "there is one
and only one" is used to indicate that exactly one object with a
certain property exists. In mathematical
logic, this sort of quantification is known
as uniqueness quantification or unique existential
quantification.

Uniqueness quantification is denoted with the
symbol "∃!". For example, the formal statement

- \exists! n \in \mathbb\,(n - 2 = 4)

## Reduction to ordinary existential and universal quantification

Uniqueness quantification can be expressed in
terms of the existential
and universal
quantifiers of predicate
logic by defining the formula ∃!x P(x) to mean

- \exists x\,(P(x) \And \forall y\,(P(y) \to x = y)).

- \exists x\,P(x) \And \forall y\, \forall z\,(P(y) \And P(z) \to y = z).

- \exists x\,\forall y\,(x = y \leftrightarrow P(x)).

## Generalizations

One generalization of uniqueness quantification
is counting
quantification. This includes both quantification of the form
"exactly k objects exist such that ..." as well as "infinitely many
objects exist such that ..." and "only finitely many object exist
such that...". The first of these forms is expressible using
ordinary quantifiers, but the latter two cannot be expressed in
ordinary first-order
logic.

## See also

uniqueness in French: Unicité
(mathématiques)

uniqueness in Dutch: Uniciteit

uniqueness in Swedish: Unik

uniqueness in Turkish: Biricik

uniqueness in Chinese: 唯一量化