# Category:Induced Mappings

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.

### I

### M

- Mapping Induced by Inverse of Injection is Surjection
- Mapping Induced on Power Set by Bijection
- Mapping Induced on Power Set by Injection is Injection
- Mapping Induced on Power Set by Inverse Relation is Mapping
- Mapping Induced on Power Set by Surjection is Surjection
- Mapping Induced on Power Set is Mapping