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{{Short description|Logical quantifier}}
{{redirect|Unique (mathematics)|other uses|Unique (disambiguation)}}
{{More sources|date=November 2024}}
In [[mathematics]] and [[logic]], the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition.<ref>{{Cite web|url=http://mathworld.wolfram.com/UniquenessTheorem.html|title=Uniqueness Theorem|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-15}}</ref> This sort of [[Quantifier (logic)|quantification]] is known as '''uniqueness quantification''' or '''unique existential quantification''', and is often denoted with the symbols "[[Existential quantification|∃]]!"<ref>{{Cite web|url=https://www.whitman.edu/mathematics/higher_math_online/section02.05.html|title=2.5 Uniqueness Arguments|website=www.whitman.edu|access-date=2019-12-15}}</ref> or "∃<sub>=1</sub>". It is defined to mean [[there exists]] an object with the given property, and [[Universal quantification|all objects]] with this property are [[Equality (mathematics)|equal]].

For example, the formal statement

: <math>\exists! n \in \mathbb{N}\,(n - 2 = 4)</math>

may be read as "there is exactly one natural number <math>n</math> such that <math>n - 2 =4</math>".

== Proving uniqueness ==

The most common technique to prove the unique existence of an object is to first prove the existence of the entity with the desired condition, and then to prove that any two such entities (say, ''<math>a</math>'' and ''<math>b</math>'') must be equal to each other (i.e. <math>a = b</math>).

For example, to show that the equation <math>x + 2 = 5</math> has exactly one solution, one would first start by establishing that at least one solution exists, namely 3; the proof of this part is simply the verification that the equation below holds:

:<math> 3 + 2 = 5. </math>

To establish the uniqueness of the solution, one would proceed by assuming that there are two solutions, namely ''<math>a</math>'' and ''<math>b</math>'', satisfying <math>x + 2 = 5</math>. That is,

:<math> a + 2 = 5\text{ and }b + 2 = 5. </math>

Then since equality is a [[transitive relation]],

:<math> a + 2 = b + 2. </math>

Subtracting 2 from both sides then yields
:<math> a = b. </math>

which completes the proof that 3 is the unique solution of <math>x + 2 = 5</math>.

In general, both existence (there exists ''at least'' one object) and uniqueness (there exists ''at most'' one object) must be proven, in order to conclude that there exists exactly one object satisfying a said condition.

An alternative way to prove uniqueness is to prove that there exists an object <math>a</math> satisfying the condition, and then to prove that every object satisfying the condition must be equal to <math>a</math>.

== Reduction to ordinary existential and universal quantification ==
Uniqueness quantification can be expressed in terms of the [[existential quantifier|existential]] and [[universal quantifier|universal]] quantifiers of [[predicate logic]], by defining the formula <math>\exists ! x P(x)</math> to mean<ref>{{Cite book |last=Kleene |first=Stephen Cole |author-link=Stephen Cole Kleene |url=https://archive.org/details/mathematicallogi0000klee/page/154 |title=Mathematical logic |date=1967 |publisher=Wiley |isbn=978-0-471-49033-3 |location=New York |pages=154 |lccn=66-26747}}</ref>

:<math>\exists x\,( P(x) \, \wedge \neg \exists y\,(P(y) \wedge y \ne x)),</math>
which is logically equivalent to
:<math>\exists x \, ( P(x) \wedge \forall y\,(P(y) \to y = x)).</math>

An equivalent definition that separates the notions of existence and uniqueness into two clauses, at the expense of brevity, is
:<math>\exists x\, P(x) \wedge \forall y\, \forall z\,[(P(y) \wedge P(z)) \to y = z].</math>

Another equivalent definition, which has the advantage of brevity, is
:<math>\exists x\,\forall y\,(P(y) \leftrightarrow y = x).</math>

== Generalizations ==
The uniqueness quantification can be generalized into [[counting quantification]] (or numerical quantification<ref>{{Cite web|url=http://persweb.wabash.edu/facstaff/helmang/phi270-1314F/phi270PDF/phi270text/phi270txt8/phi270txt83/phi270txt83(4up).pdf|title=Numerical quantification|last=Helman|first=Glen|date=August 1, 2013|website=persweb.wabash.edu|access-date=2019-12-14}}</ref>). 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 objects 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]].<ref>This is a consequence of the [[compactness theorem]].</ref>

Uniqueness depends on a notion of [[equality (mathematics)|equality]]. Loosening this to a coarser [[equivalence relation]] yields quantification of uniqueness [[up to]] that equivalence (under this framework, regular uniqueness is "uniqueness up to equality"). This is called [[essentially unique]]. For example, many concepts in [[category theory]] are defined to be unique up to [[isomorphism]].

The exclamation mark <math>!</math> can be also used as a separate quantification symbol, so <math>(\exists ! x. P(x))\leftrightarrow ((\exists x. P(x))\land (! x. P(x)))</math>, where <math>(! x. P(x)) := (\forall a \forall b. P(a)\land P(b)\rightarrow a=b)</math>. E.g. it can be safely used in the [[replacement axiom]], instead of <math>\exists !</math>.

== See also ==
* [[Essentially unique]]
* [[Extension by definition]]
* [[One-hot]]
* [[Singleton (mathematics)]]
* [[Uniqueness theorem]]

== References ==

{{reflist}}

== Bibliography ==

*{{cite book|last=Kleene|first=Stephen|title=Introduction to Metamathematics|year=1952|publisher=Ishi Press International|pages=199}}
*{{cite book|last=Andrews|first=Peter B.|title=An introduction to mathematical logic and type theory to truth through proof|year=2002|publisher=Kluwer Acad. Publ.|location=Dordrecht|isbn=1-4020-0763-9|pages=233|edition=2.}}

{{Mathematical logic}}

[[Category:Quantifier (logic)]]
[[Category:1 (number)]]
[[Category:Mathematical terminology]]
[[Category:Uniqueness theorems]]

Uniqueness quantification - Revision history

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