Definition:Direct Image Mapping/Mapping
Definition
Let $S$ and $T$ be sets.
Let $\powerset S$ and $\powerset T$ be their power sets.
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 $f$:
- $\forall X \in \powerset S: \map {f^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \map f s = t} & : X \ne \O \\ \O & : X = \O \end {cases}$
Direct Image Mapping as Set of Images of Subsets
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The direct image mapping of $f$ can be seen to be the set of images of all the subsets of the domain of $f$:
- $\forall X \subseteq S: f \sqbrk X = \map {f^\to} X$
Both approaches to this concept are used in $\mathsf{Pr} \infty \mathsf{fWiki}$.
Also known as
Some sources refer to this as the mapping induced (on the power set) by $f$.
The word defined can sometimes be seen instead of induced.
Also denoted as
The notation used here is that found in 1975: T.S. Blyth: Set Theory and Abstract Algebra.
The direct image mapping is also denoted $\powerset f$; see the covariant power set functor.
Also see
- Direct Image Mapping of Mapping is Mapping, which proves that $f^\to$ is indeed a mapping.
- Definition:Inverse Image Mapping, where the notation $f^\gets$ is used for the mapping induced by $f^{-1}$.
- Results about direct image mappings can be found here.
Generalizations
Related Concepts
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 10$: Inverses and Composites
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Functions
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections