Category:Definitions/Direct Image Mappings
Jump to navigation
Jump to search
This category contains definitions related to Direct Image Mappings.
Related results can be found in Category:Direct 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 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 $f$:
- $\forall X \in \powerset S: \map {f^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \map f s = t} & : X \ne \O \\ \O & : X = \O \end {cases}$
Pages in category "Definitions/Direct Image Mappings"
The following 7 pages are in this category, out of 7 total.