Definition:Inverse Image Mapping/Mapping
Definition
Let $S$ and $T$ be sets.
Let $\powerset S$ and $\powerset T$ be their power sets.
Let $f: S \to T$ be a mapping.
Definition 1
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 $f^{-1} \sqbrk T$ under $f$:
- $\forall Y \in \powerset T: \map {f^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \map f s = t} & : \Img f \cap Y \ne \O \\ \O & : \Img f \cap Y = \O \end {cases}$
Definition 2
The inverse image mapping of $f$ is the direct image mapping of the inverse $f^{-1}$ of $f$:
- $f^\gets = \paren {f^{-1} }^\to: \powerset T \to \powerset S$:
That is:
- $\forall Y \in \powerset T: \map {f^\gets} Y = \set {s \in S: \exists t \in Y: \map f s = t}$
Inverse Image Mapping as Set of Preimages of Subsets
The inverse image mapping of $f$ can be seen to be the set of preimages of all the subsets of the codomain of $f$.
- $\forall Y \subseteq T: f^{-1} \sqbrk Y = \map {f^\gets} Y$
Both approaches to this concept are used in $\mathsf{Pr} \infty \mathsf{fWiki}$.
Also defined as
Many authors define this concept only when $f$ is itself a mapping.
Also known as
The inverse image mapping of $f$ is also known as the preimage mapping of $f$.
Some sources refer to this as the mapping induced (on the power set) by the inverse $f^{-1}$.
Also denoted as
The notation used here is found in 1975: T.S. Blyth: Set Theory and Abstract Algebra.
The inverse image mapping can also be denoted $\map {\operatorname {\overline \PP} } f$; see the contravariant power set functor.
Also see
- Inverse Image Mapping of Mapping is Mapping, which proves that $f^\gets$ is indeed a mapping.
- Results about inverse image mappings can be found here.
Generalizations