Category:Definitions/Inverse Image Mappings
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This category contains definitions related to Inverse Image Mappings.
Related results can be found in Category: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 $f$:
- $\forall Y \in \powerset T: \map {f^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \map f s = t} & : \Img f \cap Y \ne \O \\ \O & : \Img f \cap Y = \O \end {cases}$
Pages in category "Definitions/Inverse Image Mappings"
The following 9 pages are in this category, out of 9 total.
I
- Definition:Inverse Image Mapping
- Definition:Inverse Image Mapping/Mapping
- Definition:Inverse Image Mapping/Mapping/Definition 1
- Definition:Inverse Image Mapping/Mapping/Definition 2
- Definition:Inverse Image Mapping/Relation
- Definition:Inverse Image Mapping/Relation/Definition 1
- Definition:Inverse Image Mapping/Relation/Definition 2