uniqueness n : the quality of being one of a kind; "that singularity distinguished him from all his companions" [syn: singularity]
state or quality of being unique or one-of-a-kind
- Czech: jedinečnost
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)).
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.
uniqueness in French: Unicité (mathématiques)
uniqueness in Dutch: Uniciteit
uniqueness in Swedish: Unik
uniqueness in Turkish: Biricik
uniqueness in Chinese: 唯一量化