Definition:Induced Mapping

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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 $T$:

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

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.

This is sometimes called the direct image of $X$ under $\mathcal R$.

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

Some authors, for example T.S. Blyth: Set Theory and Abstract Algebra (1975), 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 $g$.