# 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 $P \left({x}\right)$ 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:

## 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 $x$

- there exists
- and:
- there exists
**at least**one $x$.

- there exists

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.

## Also see

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.): $\S 4.26$ - 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 6.10$