Definition:Inverse Image Mapping/Relation

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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

  • Results about inverse image mappings can be found here.

Special cases

Related Concepts