> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/220-relation-theory/relation-properties/uniqueness-properties.md).

# Uniqueness properties of relations

Uniqueness properties:

* Injective, left-unique
* Functional, right-unique, right-definite, univalent
* **Injective** or *left-unique*: `∀xz ∈ X. ∀y ∈ Y. xRy ⋀ zRy -> x = z`

For such a relation, `{Y}` is called a *primary key* of R.

For example, the green and blue binary relations in the diagram are injective, but the red one is not (as it relates both −1 and 1 to 1), nor the black one (as it relates both −1 and 1 to 0).

* **Functional** or right-unique, right-definite or univalent:

for all � ∈ � x\in X and all � , � ∈ � , {\displaystyle y,z\in Y,} if xRy and xRz then y = z. Such a binary relation is called a *partial function*. For such a relation, { � } {\displaystyle {X}} is called a primary key of R.

For example, the red and green binary relations in the diagram are functional, but the blue one is not (as it relates 1 to both −1 and 1), nor the black one (as it relates 0 to both −1 and 1).

* One-to-one: injective and functional. For example, the green binary relation in the diagram is one-to-one, but the red, blue and black ones are not.
* One-to-many: injective and not functional. For example, the blue binary relation in the diagram is one-to-many, but the red, green and black ones are not.
* Many-to-one: functional and not injective. For example, the red binary relation in the diagram is many-to-one, but the green, blue and black ones are not.
* Many-to-many: not injective nor functional. For example, the black binary relation in the diagram is many-to-many, but the red, green and blue ones are not.