# Uniqueness quantification

Uniqueness quantification

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 often denoted with the symbols "∃!" or ∃=1". For example, the formal statement

$\exists! n \in \mathbb{N}\,(n - 2 = 4)$

may be read aloud as "there is exactly one natural number n such that n - 2 = 4".

## Proving uniqueness

Proving uniqueness turns out to be mostly easier than that of existence or expressibility. The most common technique to proving uniqueness is to assume there exists two quantities (say, a and b) that satisfies the condition given, and then logically deducing their equality, i.e. a = b.

As a simple high school example, to show x + 2 = 5 has only one solution, we assume there are two solutions first, namely, a and b, satisfying x + 2 = 5. Thus

$a + 2 = 5\text{ and }b + 2 = 5. \,$

By transitivity of equality,

$a + 2 = b + 2. \,$

By cancellation,

$a = b. \,$

This simple example shows how a proof of uniqueness is done, the end result being the equality of the two quantities that satisfy the condition. We must say, however, that existence/expressibility must be proven before uniqueness, or else we cannot even assume the existence of those two quantities to begin with.

## 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

$\exists x P(x) \wedge \neg \exists x,y \, ( P(x) \wedge P(y) \wedge (x \neq y))$

where an equivalence is:

$\exists x P(x) \wedge \forall y\,(P(y) \to x = y).$

An equivalent definition that has the virtue of separating 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 with the advantage of brevity is

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

## 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 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.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Quantification — has two distinct meanings.In mathematics and empirical science, it refers to human acts, known as counting and measuring that map human sense observations and experiences into members of some set of numbers. Quantification in this sense is… …   Wikipedia

• Counting quantification — A counting quantifier is a mathematical term for a quantifier of the form there exists at least k elements that satisfy property X . In first order logic with equality, counting quantifiers can be defined in terms of ordinary quantifiers, so in… …   Wikipedia

• First-order logic — is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic (a less… …   Wikipedia

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# Uniqueness quantification

Uniqueness quantification

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 often denoted with the symbols "∃!" or ∃=1". For example, the formal statement

$\exists! n \in \mathbb{N}\,(n - 2 = 4)$

may be read aloud as "there is exactly one natural number n such that n - 2 = 4".

## Proving uniqueness

Proving uniqueness turns out to be mostly easier than that of existence or expressibility. The most common technique to proving uniqueness is to assume there exists two quantities (say, a and b) that satisfies the condition given, and then logically deducing their equality, i.e. a = b.

As a simple high school example, to show x + 2 = 5 has only one solution, we assume there are two solutions first, namely, a and b, satisfying x + 2 = 5. Thus

$a + 2 = 5\text{ and }b + 2 = 5. \,$

By transitivity of equality,

$a + 2 = b + 2. \,$

By cancellation,

$a = b. \,$

This simple example shows how a proof of uniqueness is done, the end result being the equality of the two quantities that satisfy the condition. We must say, however, that existence/expressibility must be proven before uniqueness, or else we cannot even assume the existence of those two quantities to begin with.

## 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

$\exists x P(x) \wedge \neg \exists x,y \, ( P(x) \wedge P(y) \wedge (x \neq y))$

where an equivalence is:

$\exists x P(x) \wedge \forall y\,(P(y) \to x = y).$

An equivalent definition that has the virtue of separating 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 with the advantage of brevity is

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

## 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 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.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Quantification — has two distinct meanings.In mathematics and empirical science, it refers to human acts, known as counting and measuring that map human sense observations and experiences into members of some set of numbers. Quantification in this sense is… …   Wikipedia

• Counting quantification — A counting quantifier is a mathematical term for a quantifier of the form there exists at least k elements that satisfy property X . In first order logic with equality, counting quantifiers can be defined in terms of ordinary quantifiers, so in… …   Wikipedia

• First-order logic — is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic (a less… …   Wikipedia

• List of mathematics articles (U) — NOTOC U U duality U quadratic distribution U statistic UCT Mathematics Competition Ugly duckling theorem Ulam numbers Ulam spiral Ultraconnected space Ultrafilter Ultrafinitism Ultrahyperbolic wave equation Ultralimit Ultrametric space… …   Wikipedia

• List of topics in logic — This is a list of topics in logic.See also: List of mathematical logic topicsAlphabetical listAAbacus logic Abduction (logic) Abductive validation Affine logic Affirming the antecedent Affirming the consequent Antecedent Antinomy Argument form… …   Wikipedia

• Well-defined — In mathematics, the term well defined is used to specify that a certain concept or object (a function, a property, a relation, etc.) is defined in a mathematical or logical way using a set of base axioms in an entirely unambiguous way and… …   Wikipedia

• Well-definition — For other uses, see Definition (disambiguation). In mathematics, well definition is a mathematical or logical definition of a certain concept or object (a function, a property, a relation, etc.) which uses a set of base axioms in an entirely… …   Wikipedia

• ! (disambiguation) — ! is a punctuation mark called an exclamation mark, exclamation point, ecphoneme, or bang. ! may also refer to: Contents 1 Mathematics and computers 2 Music 3 …   Wikipedia

• List of basic discrete mathematics topics — Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally , in the sense of not supporting or requiring the notion of continuity. Most, if not all, of the objects studied in finite… …   Wikipedia

• Definite description — A definite description is a denoting phrase in the form of the X where X is a noun phrase or a singular common noun. The definite description is proper if X applies to a unique individual or object. For example: the first person in space and the… …   Wikipedia
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The provided text does not contain any mathematical questions or calculations, but it does discuss mathematical concepts, specifically the concept of uniqueness quantification in mathematics and logic. The text explains the notation and methods used to prove uniqueness, but does not pose or answer any specific mathematical questions.

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