# Definition:Composition of Mappings/Binary Operation

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## Definition

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 $\sqbrk {S \to S}$.

## Sources

- 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.4$ Example $3$

- 2011: Robert G. Bartle and Donald R. Sherbert:
*Introduction to Real Analysis*(4th ed.) ... (previous) ... (next): $\S 1.1$: Sets and Functions