# Category:Induced Mappings It has been suggested that this page or section be renamed. One may discuss this suggestion on the talk page.

This category contains results about Induced Mappings.
Definitions specific to this category can be found in Definitions/Induced Mappings.

Let $S$ and $T$ be sets.

Let $\powerset S$ and $\powerset T$ be their power sets.

### Relation

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 $\mathcal R$:

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

### Mapping

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 $f$:

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

## Subcategories

This category has only the following subcategory.

## Pages in category "Induced Mappings"

The following 10 pages are in this category, out of 10 total.