Definition:Composition of Mappings/Warning

Definition

Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that:

$\Dom {f_2} \ne \Cdm {f_1}$

where $\Dom {f_2}$ and $\Cdm {f_1}$ denote domain and codomain respectively.

Then the composite mapping $f_2 \circ f_1$ is not defined.

Compare with the definition of composition of relations in the context of 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): Chapter $\text I$: Algebraic Structures: $\S 5$: Composites and Inverses of Functions
• 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

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