Category:Definitions/Induced Mappings

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This category contains definitions related to Induced Mappings.
Related results can be found in Category:Induced Mappings.


The mapping induced on a powerset can be either of two concepts:


Direct Image Mapping

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

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


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 $f$:

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


Inverse Image Mapping

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 $f$:

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

Subcategories

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

Pages in category "Definitions/Induced Mappings"

The following 3 pages are in this category, out of 3 total.