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 $S \times T$.

Definition 2
Note that:
 * $\mathcal R^\gets \paren T = \Preimg {\mathcal R}$

where $\Preimg {\mathcal R}$ is the preimage of $\mathcal R$.

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 or the induced mapping on power sets by the inverse $\mathcal R^{-1}$.

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

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

Also see

 * Equivalence of Definitions of Inverse Image Mapping of Relation


 * Mapping Induced on Power Set is Mapping, which proves that $\mathcal R^\to$, and so therefore $\mathcal R^\gets = \paren {\mathcal R^{-1} }^\to$, is indeed a mapping for any relation $\mathcal R$.

Special cases

 * Definition:Inverse Image Mapping of Mapping