Definition:Extension/Mapping

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 $f_2 \restriction_{X \times Y} = f_1$.

Then $f_2$ extends or is an extension of $f_1$.

Also see

• Restriction of a Mapping

• Extension of a Relation
• Extension of an Operation

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 8$: Functions
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 8$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.4$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.4$