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

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

Contents

• 1 Proving uniqueness
• 2 Reduction to ordinary existential and universal quantification
• 3 Generalizations
• 4 See also

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

By transitivity of equality,

By cancellation,

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.

Adjective

unique (epicene, plural uniques)

1. unique
2. only

Related terms

• monnaie unique
• unicité
• uniquement

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