# Category:Inverse Image Mappings

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This category contains results about **Inverse Image Mappings**.

Definitions specific to this category can be found in Definitions/Inverse Image Mappings.

Let $S$ and $T$ be sets.

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

### Relation

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

The **inverse image mapping** of $\RR$ is the mapping $\RR^\gets: \powerset T \to \powerset S$ that sends a subset $Y \subseteq T$ to its preimage $\map {\RR^{-1} } Y$ under $\RR$:

- $\forall Y \in \powerset T: \map {\RR^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \tuple {t, s} \in \RR^{-1} } & : \Img \RR \cap Y \ne \O \\ \O & : \Img \RR \cap Y = \O \end {cases}$

### Mapping

Let $f: S \to T$ be a mapping.

The **inverse image mapping** of $f$ is the mapping $f^\gets: \powerset T \to \powerset S$ that sends a subset $Y \subseteq T$ to its preimage $f^{-1} \paren T$ under $f$:

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

## Also see

## Pages in category "Inverse Image Mappings"

The following 14 pages are in this category, out of 14 total.

### D

### E

### I

- Inverse Image Mapping Induced by Projection
- Inverse Image Mapping is Mapping
- Inverse Image Mapping of Codomain is Preimage Set of Mapping
- Inverse Image Mapping of Codomain is Preimage Set of Relation
- Inverse Image Mapping of Injection is Surjection
- Inverse Image Mapping of Mapping is Mapping
- Inverse Image Mapping of Relation is Mapping
- Inverse Image of Direct Image of Inverse Image equals Inverse Image Mapping
- Inverse of Direct Image Mapping does not necessarily equal Inverse Image Mapping