# 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.

## Also see

- Mapping Induced on Power Set is Mapping, which proves that $f^\to$ is indeed a mapping.
- Definition:Inverse Image Mapping, where the notation $f^\gets$ is used for the
**mapping induced by $f^{-1}$**.

### Generalization

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 10$: Inverses and Composites - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{I}$: Functions - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.2$: Homomorphisms - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 5$