# Definition:Direct Image Mapping/Mapping

## Definition

Let $S$ and $T$ be sets.

Let $\powerset S$ and $\powerset T$ 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 $f^\to: \powerset S \to \powerset T$ that sends a subset $X \subseteq S$ to its image under $f$:

- $\forall X \in \powerset S: \map {f^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \map f s = t} & : X \ne \O \\ \O & : X = \O \end {cases}$

### Direct Image Mapping as Set of Images of Subsets

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The **direct image mapping** of $f$ can be seen to be the set of images of all the subsets of the domain of $f$:

- $\forall X \subseteq S: f \sqbrk X = \map {f^\to} X$

Both approaches to this concept are used in $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also known as

Some sources refer to this as the **mapping induced (on the power set) by $f$**.

The word **defined** can sometimes be seen instead of **induced**.

## Also denoted as

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

The **direct image mapping** is also denoted $\powerset f$; see the covariant power set functor.

## Also see

- Direct Image Mapping of Mapping 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 $f^{-1}$**.

- Results about
**direct image mappings**can be found**here**.

### Generalizations

### Related Concepts

## 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): Chapter $\text{I}$: Sets and Functions: 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): $\S 5$. Induced mappings; composition; injections; surjections; bijections