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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, whe ...
!" or "∃=1". For example, the formal statement : \exists! n \in \mathbb\,(n - 2 = 4) may be read as "there is exactly one natural number n such that n - 2 =4".


Proving uniqueness

The most common technique to prove the unique existence of a certain object is to first prove the existence of the entity with the desired condition, and then to prove that any two such entities (say, ''a'' and ''b'') must be equal to each other (i.e. a = b). For example, to show that the equation x + 2 = 5 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: : 3 + 2 = 5. To establish the uniqueness of the solution, one would then proceed by assuming that there are two solutions, namely ''a'' and ''b'', satisfying x + 2 = 5. That is, : a + 2 = 5\textb + 2 = 5. By transitivity of equality, : a + 2 = b + 2. Subtracting 2 from both sides then yields : a = b. which completes the proof that 3 is the unique solution of x + 2 = 5. 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 a satisfying the condition, and then to prove that every object satisfying the condition must be equal to a.


Reduction to ordinary existential and universal quantification

Uniqueness quantification can be expressed in terms of the
existential Existentialism ( ) is a form of philosophical inquiry that explores the problem of human existence and centers on human thinking, feeling, and acting. Existentialist thinkers frequently explore issues related to the meaning, purpose, and valu ...
and
universal Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company ** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal ** Universal TV, a t ...
quantifiers of predicate logic, by defining the formula \exists ! x P(x) to mean :\exists x\,( P(x) \, \wedge \neg \exists y\,(P(y) \wedge y \ne x)), which is logically equivalent to :\exists x \, ( P(x) \wedge \forall y\,(P(y) \to y = x)). An equivalent definition that separates the notions of existence and uniqueness into two clauses, at the expense of brevity, is :\exists x\, P(x) \wedge \forall y\, \forall z\, P(y) \wedge P(z)) \to y = z Another equivalent definition, which has the advantage of brevity, is :\exists x\,\forall y\,(P(y) \leftrightarrow y = x).


Generalizations

The uniqueness quantification can be generalized into counting quantification (or numerical 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 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 First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
.This is a consequence of the
compactness theorem In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally ...
.
Uniqueness depends on a notion of equality. Loosening this to some coarser
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
yields quantification of uniqueness
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
that equivalence (under this framework, regular uniqueness is "uniqueness up to equality"). For example, many concepts in
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
are defined to be unique up to
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. The exclamation mark ! can be also used as a separate quantification symbol, so (\exists ! x. P(x))\leftrightarrow ((\exists x. P(x))\land (! x. P(x))), where (! x. P(x)) := (\forall a \forall b. P(a)\land P(b)\rightarrow a=b). E.g. it can be safely used in the
replacement axiom In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite ...
, instead of \exists !.


See also

* Essentially unique * One-hot *
Singleton (mathematics) In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...
*
Uniqueness theorem In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems ...


References


Bibliography

* * {{Mathematical logic Quantifier (logic) 1 (number) Mathematical terminology Uniqueness theorems