Properties
Composition of relations is associative.
The inverse relation of S ∘ R is (S ∘ R)-1 = R−1 ∘ S−1. This property makes the set of all binary relations on a set a semigroup with involution.
The compose of (partial) functions (i.e. functional relations) is again a (partial) function.
If R and S are injective, then S ∘ R is injective, which conversely implies only the injectivity of R.
If R and S are surjective, then S ∘ R is surjective, which conversely implies only the surjectivity of S.
The set of binary relations on a set X (i.e. relations from X to X) together with (left or right) relation composition forms a monoid with zero, where the identity map on X is the neutral element, and the empty set is the zero element.
Read more about this topic: Composition Of Relations
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)