In
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 premise ...
, 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 "
∃!" or "∃
=1". For example, the formal statement
:
may be read as "there is exactly one natural number
such that
".
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, ''
'' and ''
'') must be equal to each other (i.e.
).
For example, to show that the equation
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:
:
To establish the uniqueness of the solution, one would then proceed by assuming that there are two solutions, namely ''
'' and ''
'', satisfying
. That is,
:
By
transitivity of equality,
:
Subtracting 2 from both sides then yields
:
which completes the proof that 3 is the unique solution of
.
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
satisfying the condition, and then to prove that every object satisfying the condition must be equal to
.
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 quantifiers of
predicate 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 ...
, by defining the formula
to mean
:
which is logically equivalent to
:
An equivalent definition that separates the notions of existence and uniqueness into two clauses, at the expense of brevity, is
:
Another equivalent definition, which has the advantage of brevity, is
:
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
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elit ...
. Loosening this to some coarser
equivalence relation yields quantification of uniqueness
up to that equivalence (under this framework, regular uniqueness is "uniqueness up to equality"). For example, many concepts in
category theory 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
, where
. E.g. it can be safely used in the
replacement axiom, instead of
.
See also
*
Essentially unique In mathematics, the term essentially unique is used to describe a weaker form of uniqueness, where an object satisfying a property is "unique" only in the sense that all objects satisfying the property are equivalent to each other. The notion of ess ...
*
One-hot
In digital circuits and machine learning, a one-hot is a group of bits among which the legal combinations of values are only those with a single high (1) bit and all the others low (0). A similar implementation in which all bits are '1' except ...
*
Singleton (mathematics)
*
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