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

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

The most common technique to proving unique existence is to first prove existence of entity with the desired condition; then, to assume there exist two entities (say, a and b) that both satisfy the condition, and logically deduce their equality, i.e. a = b.

As a simple high school example, to show x + 2 = 5 has exactly one solution, we first show by demonstration that at least one solution exists, namely 3; the proof of this part is simply the calculation

We now assume that there are two solutions, 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.

Both existence and uniqueness must be proven, in order to conclude that there exists exactly one solution.

An alternative way to prove uniqueness is to prove there exists a value ${\displaystyle a}$ satisfying the condition, and then proving that, for all ${\displaystyle x}$, the condition for ${\displaystyle x}$ implies ${\displaystyle x=a}$.

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