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.