# Definition:Direct Image Mapping/Relation

## Contents

## Definition

Let $S$ and $T$ be sets.

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

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

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

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

Note that:

- $\mathcal R^\to \left({S}\right) = \operatorname{Im} \left({\mathcal R}\right)$

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

## Also defined as

Many authors define this concept only when $\mathcal R$ is itself a mapping.

## Also known as

The **direct image mapping** of $\mathcal R$ is also known as the **mapping induced on power sets by $\mathcal R$** or the **mapping defined by $\mathcal R$**.

## Also denoted as

The notation used here is derived from similar notation for the mapping induced by a mapping found in 1975: T.S. Blyth: *Set Theory and Abstract Algebra*.

## Also see

- Mapping Induced on Power Set is Mapping, which proves that $\mathcal R^\to$ is indeed a mapping.

### Special cases

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{I}$: Problem $\text{AA}$