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
Famous quotes containing the words special, types and/or relations:
“In England and America a beard usually means that its owner would rather be considered venerable than virile; on the continent of Europe it often means that its owner makes a special claim to virility.”
—Rebecca West (1892–1983)
“The bourgeoisie loves so-called “positive” types and novels with happy endings since they lull one into thinking that it is fine to simultaneously acquire capital and maintain one’s innocence, to be a beast and still be happy.”
—Anton Pavlovich Chekhov (1860–1904)
“All social rules and all relations between individuals are eroded by a cash economy, avarice drags Pluto himself out of the bowels of the earth.”
—Karl Marx (1818–1883)