Binary Relation - Special Types of Binary Relations

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.

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