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

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 $$g$$.