Definition:Extension of Mapping
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This page is about extensions of mappings. For other uses, see Extension.
Definition
As a mapping is, by definition, also a relation, the definition of an extension of a mapping is the same as that for an extension of a relation:
Let:
- $f_1 \subseteq X \times Y$ be a mapping on $X \times Y$
- $f_2 \subseteq S \times T$ be a mapping on $S \times T$
- $X \subseteq S$
- $Y \subseteq T$
- $f_2 \restriction_{X \times Y}$ be the restriction of $f_2$ to $X \times Y$.
Let $f_2 \restriction_{X \times Y} = f_1$.
That is, let $f_1$ be a subset of $f_2$.
Then $f_2$ extends or is an extension of $f_1$.
Examples
Extension of Square Function on Natural Numbers
Let $f: \N \to \N$ be the mapping defined as:
- $\forall n \in \N: \map f n = n^2$
Let $h: \R \to \R$ be the mapping defined as:
- $\forall x \in \R: \map h x = x^2$
Then $h$ is a extension of $f$.
Also see
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Functions
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 8$: Functions
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets
- 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: Restrictions and Extensions
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 9$: Inverse Functions, Extensions, and Restrictions
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.4$: Composition and Restriction
- 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Power Set
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Functions
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 1$ A few preliminaries