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In mathematics, the composition of binary relations is a concept of forming a new relation S o R from two given relations R and S, having as its most well-known special case the composition of functions. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ...
In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...
Definition
If and are two binary relations, then their compose is the relation - defined by
where, as usual, .
In other words, S o R is the relation from X to Z whose graph is
- .
In particular fields, authors might denote by R o S what is defined here to be S o R. The convention chosen here is such that function composition (with the usual notation) is obtained as a special case, when R and S are functional relations. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...
Properties Composition of relations is associative. In mathematics, associativity is a property that a binary operation can have. ...
The inverse relation of S o R is (S o R)-1 = R-1 o S-1. In logic and mathematics, the inverse relation of a binary relation is the binary relation defined by . ...
The compose of (partial) functions (i.e. functional relations) is again a (partial) function. In mathematics, a partial function is a relation that associates each element of a set (sometimes called its domain) with at most one element of another (possibly the same) set, called the codomain. ...
If R and S are injective, then S o R is injective, which conversely implies only the injectivity of R. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
If R and S are surjective, then S o R is surjective, which conversely implies only the surjectivity of S. In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...
The binary relations on a set X (i.e. relations from X to X) form a monoid for composition, with the identity map on X as neutral element. In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...
In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...
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