# Category:Direct Image Mappings

This category contains results about Direct Image Mappings.

Definitions specific to this category can be found in Definitions/Direct Image 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 $\mathcal R^\to: \powerset S \to \powerset T$ that sends a subset $X \subseteq T$ to its image under $\mathcal R$:

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

### Mapping

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

The **direct image mapping** of $f$ is the mapping $f^\to: \powerset S \to \powerset T$ that sends a subset $X \subseteq S$ to its image under $f$:

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

## Also see

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Direct Image Mappings"

The following 20 pages are in this category, out of 20 total.

### C

### D

- Direct Image Mapping is Bijection iff Mapping is Bijection
- Direct Image Mapping is Mapping
- Direct Image Mapping of Domain is Image Set of Mapping
- Direct Image Mapping of Domain is Image Set of Relation
- Direct Image Mapping of Injection is Injection
- Direct Image Mapping of Left-Total Relation is Empty iff Argument is Empty
- Direct Image Mapping of Mapping is Empty iff Argument is Empty
- Direct Image Mapping of Mapping is Mapping
- Direct Image Mapping of Relation is Mapping
- Direct Image Mapping of Surjection is Surjection
- Direct Image of Intersection with Inverse Image
- Direct Image of Inverse Image of Direct Image equals Direct Image Mapping