Definition:Inverse Image Mapping/Relation

Definition
Let $S$ and $T$ be sets.

Let $\powerset S$ and $\powerset T$ be their power sets. Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$.

Inverse Image Mapping as Set of Preimages of Subsets
The inverse image mapping of $\mathcal R$ can be seen to be the set of preimages of all the subsets of the codomain of $\mathcal R$.


 * $\forall Y \subseteq T: \mathcal R^{-1} \sqbrk Y = \map {\mathcal R^\gets} Y$

Both approaches to this concept are used in.

Also defined as
Many authors define this concept only when $\mathcal R$ is itself a mapping.

Also known as
This inverse image mapping of $\mathcal R$ is also known as the preimage mapping of $\mathcal R$.

Some sources refer to this as the mapping induced (on the power set) by the inverse $\mathcal R^{-1}$.

Also denoted as
The notation used here is derived from similar notation for the inverse image mapping of a mapping found in.

The inverse image mapping can also be denoted $\map {\operatorname {\overline {\mathcal P} } } {\mathcal R}$; see the contravariant power set functor.

Also see

 * Equivalence of Definitions of Inverse Image Mapping of Relation


 * Inverse Image Mapping of Codomain is Preimage Set of Relation


 * Inverse Image Mapping of Relation is Mapping, which shows that $\mathcal R^\gets$ is indeed a mapping for any relation $\mathcal R$.

Special cases

 * Definition:Inverse Image Mapping of Mapping

Generalizations

 * Definition:Preimage of Relation

Related Concepts

 * Definition:Direct Image Mapping of Relation
 * Definition:Direct Image Mapping of Mapping


 * Definition:Image of Subset under Relation
 * Definition:Image of Subset under Mapping