\begin{definition}[Definition:Composition of Mappings/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 $\sqbrk {S \to S}$.
\end{definition}