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:


 * $f_2 \circ f_1 := \set {\tuple {x, z} \in S_1 \times S_3: \exists y \in S_2: \tuple {x, y} \in f_1 \land \tuple {y, z} \in f_2}$

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


 * $\forall x \in S_1: \paren {f_2 \circ f_1} \paren x := f_2 \paren {f_1 \paren x}$


 * CompositeMapping.png

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 $\sqbrk {S \to S}$ be the set of all mappings from a set $S$ to itself.

Then the concept of composite mapping defines a binary operation on $\sqbrk {S \to S}$:


 * $\forall f, g \in \sqbrk {S \to S}: g \circ f = \set {\tuple {s, t}: s \in S, \tuple {f \paren s, t} \in g} \in \sqbrk {S \to S}$

Thus, for every pair $\tuple {f, g}$ of mappings in $\sqbrk {S \to S}$, the composition $g \circ f$ is another element of $\sqbrk {S \to S}$.

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 $f_1; f_2$ or $f_1 f_2$ (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 (according to some sources) makes set theorists wince, as it is technically defined as a function on the codomain of a function.

Also see

 * Composite Mapping is Mapping


 * Domain of Composite Mapping
 * Image of Composite Mapping