Definition:Direct Image Mapping/Mapping

Definition

Let $S$ and $T$ be sets.

Let $\mathcal P(S)$ and $\mathcal P(T)$ be their power sets.

Let $f \subseteq S \times T$ be a mapping from $S$ to $T$.

The direct image mapping of $f$ is the mapping $f^\to: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$ that sends a subset $X \subseteq S$ to its image under $f$:

$\forall X \in \mathcal P \left({S}\right): f^\to \left({X}\right) = \left\{ {t \in T: \exists s \in X: f \left({s}\right) = t}\right\}$

Note that:

$f^\to \left({S}\right) = \operatorname{Im} \left({f}\right)$

where $\operatorname{Im} \left({f}\right)$ is the image set of $f$.

Also known as

The direct image mapping is also known as the mapping induced on power sets by $f$ or the mapping defined by $f$. The latter can be confused with the inverse image mapping.

Also denoted as

The notation used here is that found in 1975: T.S. Blyth: Set Theory and Abstract Algebra.

Many sources use the same notation for the induced mapping as for the mapping itself, but this can cause confusion.

The direct image mapping is also denoted $\mathcal P \left({f}\right)$; see the covariant power set functor.

Some sources use $f_g$ to denote what $\mathsf{Pr} \infty \mathsf{fWiki}$ denotes as $g^\to$, but this is confusing and is to be avoided.