Definition:Direct Image Mapping

Definition
Let $$\mathcal{R} \subseteq S \times T$$ be a relation on $$S \times T$$.

Then $$\mathcal{R}$$ defines (or induces) a mapping from the power set of $$S$$ to the power set of $$T$$:


 * $$f_{\mathcal{R}}: \mathcal{P} \left({S}\right) \to \mathcal{P} \left({T}\right): f_{\mathcal{R}} \left({X}\right) = \mathcal{R} \left({X}\right)$$

This is sometimes called the direct image of $$X$$ under $$\mathcal{R}$$. See the definition of the image of a subset.

Note that:
 * $$f_{\mathcal{R}} \left({S}\right) = \operatorname{Im} \left({f}\right)$$

That this is a mapping is proved here.

Comment
Many authors only bother to define this concept when $$\mathcal{R}$$ is itself a mapping, say $$g$$.

Some authors, for example, use $$g^\to$$ for what we would call $$f_{g}$$.

Similarly, $$g^\gets$$ is used for $$f_{g^{-1}}$$, where $$g^{-1}$$ is the inverse of $$g$$.