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:
“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)
“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)