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