# Definition:Composition of Mappings/Warning

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## 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 $f_1$.

If $\Dom {f_2} \ne \Cdm {f_1}$, then the **composite mapping** $f_2 \circ f_1$ is **not defined**.

This definition is directly analogous to that of composition of relations owing to the fact that a mapping is a special kind of relation.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.4$. Product of mappings: Example $49$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 5$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections:*Remark $1$* - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 24$: Composition of Mappings - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions

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