Special Types of Binary Relations
Some important types of binary relations R between X and Y are listed below.
Uniqueness properties:
- injective (also called left-unique): for all x and z in X and y in Y it holds that if xRy and zRy then x = z.
- functional (also called right-unique or right-definite): for all x in X, and y and z in Y it holds that if xRy and xRz then y = z; such a binary relation is called a partial function.
- one-to-one (also written 1-to-1): injective and functional.
Totality properties:
- left-total: for all x in X there exists a y in Y such that xRy (this property, although sometimes also referred to as total, is different from the definition of total in the next section).
- surjective (also called right-total): for all y in Y there exists an x in X such that xRy.
Uniqueness and totality properties:
- A function: a relation that is functional and left-total.
- A bijection: a one-to-one correspondence; such a relation is a function and is said to be bijective.
Read more about this topic: Binary Relation
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