# Definition:Unique

## Definition

Suppose $A$ and $B$ are two objects whose definition is in terms of a given set of properties.

If it can be demonstrated that, in order for both $A$ and $B$ to fulfil those properties, it is necessary for $A$ to be equal to $B$, then $A$ (and indeed $B$) is unique.

Equivalently, there is one and only one, or exactly one, such object.

Thus, intuitively, an object is unique if there is precisely one such object.

### Unique Existential Quantifier

In the language of predicate logic, uniqueness can be defined as follows:

Let $\map P x$ be a propositional function and let $x$ and $y$ be objects.

The symbol $\exists !$ denotes the existence of a unique object fulfilling a particular condition.

$\exists ! x: \map P x$

means:

There exists exactly one object $x$ such that $\map P x$ holds

or:

There exists one and only one $x$ such that $\map P x$ holds.

Formally:

$\exists ! x: \map P x \dashv \vdash \exists x: \map P x \land \forall y: \map P y \implies x = y$

In natural language, this means:

There exists exactly one $x$ with the property $P$
is logically equivalent to:
There exists an $x$ such that $x$ has the property $P$, and for every $y$, $y$ has the property $P$ only if $x$ and $y$ are the same object.

## Also known as

Uniqueness can also be defined as:

An object $x$ is unique (in a given context) if and only if:
there exists at most one $x$
and:
there exists at least one $x$.

Thus the phrase at most and at least one can occasionally be seen to mean unique.

Such a definition can be a useful technique for proving uniqueness.