Definition:Composition of Mappings

Definition
Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$.

Then the composite of $f_1$ and $f_2$ is defined and denoted as:


 * $f_2 \circ f_1 := \left\{{\left({x, z}\right) \in S_1 \times S_3: \exists y \in S_2: \left({x, y}\right) \in f_1 \land \left({y, z}\right) \in f_2}\right\}$

That is, the composite mapping $f_2 \circ f_1$ is defined as:


 * $\forall x \in S_1: \left({f_2 \circ f_1}\right) \left({x}\right) := f_2 \left({f_1 \left({x}\right)}\right)$


 * CompositeMapping.png

Domain and Codomain
From Domain of Composite Relation, the domain of $f_2 \circ f_1$ is the domain of $f_1$:
 * $\operatorname{Dom} \left({f_2 \circ f_1}\right) = \operatorname{Dom} \left({f_1}\right)$

From Codomain of Composite Relation, the codomain of $f_2 \circ f_1$ is the codomain of $f_2$:
 * $\operatorname{Cdm} \left({f_2 \circ f_1}\right) = \operatorname{Cdm} \left({f_2}\right)$

These follow because, by definition, a mapping is a special type of relation.

Commutative Diagram
The concept of composition of mappings can be illustrated by means of a commutative diagram.

This example illustrates the specific example cited here:


 * $\begin{xy}\xymatrix@+1em{

S_1 \ar[r]^*+{f_1} \ar[rd]_*[l]+{f_2 \mathop \circ f_1} & S_2 \ar[d]^*+{f_2}

\\ & S_3 }\end{xy}$

Composition as a Binary Operation
Let $\left[{S \to S}\right]$ be the set of all mappings from a set $S$ to itself.

Then the concept of composite mapping defines a binary operation on $\left[{S \to S}\right]$:


 * $\forall f, g \in \left[{S \to S}\right]: g \circ f = \left\{{\left({s, t}\right): s \in S, \left({f \left({s}\right), t}\right) \in g}\right\} \in \left[{S \to S}\right]$

Thus, for every pair $\left({f, g}\right)$ of mappings in $\left[{S \to S}\right]$, the composition $g \circ f$ is another element of $\left[{S \to S}\right]$.

Also known as
Some authors write $f_2 \circ f_1$ as $f_2 f_1$.

Others, particularly in books having ties with computer science, write $f_1; f_2$ (note the reversal of order), which is read as (apply) $f_1$, then $f_2$.

Some sources call $f_2 \circ f_1$ the resultant of $f_1$ and $f_2$ or the product of $f_1$ and $f_2$.

In the context of analysis, this is often found referred to as a function of a function, which (apparently) makes set theorists wince.

Also see

 * Composite Mapping is Mapping