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

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