Definition:Composition of Mappings/Definition 1
Definition
Let $S_1$, $S_2$ and $S_3$ be sets.
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 $f_1$.
The composite mapping $f_2 \circ f_1$ is defined as:
- $\forall x \in S_1: \map {\paren {f_2 \circ f_1} } x := \map {f_2} {\map {f_1} x}$
Commutative Diagram
The concept of composition of mappings can be illustrated by means of a commutative diagram.
This diagram illustrates the specific example of $f_2 \circ f_1$:
- $\begin{xy}\[email protected]+1em{ S_1 \ar[r]^*+{f_1} \[email protected]{-->}[rd]_*[l]+{f_2 \mathop \circ f_1} & S_2 \ar[d]^*+{f_2} \\ & S_3 }\end{xy}$
Warning
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.
Also known as
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.
Some sources call $f_2 \circ f_1$ the resultant of $f_1$ and $f_2$ or the product of $f_1$ and $f_2$.
Some authors write $f_2 \circ f_1$ as $f_2 f_1$.
Some use the notation $f_2 \cdot f_1$ or $f_2 . f_1$.
Some use the notation $f_2 \bigcirc f_1$.
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 $f_2$.
Also see
- Results about composite mappings can be found here.
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 2$: Product sets, mappings
- 1959: E.M. Patterson: Topology (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 9$. Functions
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 10$: Inverses and Composites
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.3: \ 4$
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.4$. Product of mappings
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 5$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.4$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Mappings: $\S 16$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.14$: Composition of Functions: Definition $14.1$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Composition of functions
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- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 8$: Composition of Functions and Diagrams
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- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.4$: Functions
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- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 24$: Composition of Mappings
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.6$: Functions
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- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.4$: Composition and Restriction
- 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions: $10$
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Definition $2.1.3$
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Functions
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (next): $\S 1.2$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: composition
- 2011: Robert G. Bartle and Donald R. Sherbert: Introduction to Real Analysis (4th ed.) ... (previous) ... (next): $\S 1.1$: Sets and Functions