Definition:Preimage/Relation/Subset
Definition
Let $\RR \subseteq S \times T$ be a relation.
Let $\RR^{-1} \subseteq T \times S$ be the inverse relation to $\RR$, defined as:
- $\RR^{-1} = \set {\tuple {t, s}: \tuple {s, t} \in \RR}$
Let $Y \subseteq T$.
The preimage of $Y$ under $\RR$ is defined as:
- $\RR^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \tuple {s, t} \in \RR}$
That is, the preimage of $Y$ under $\RR$ is the image of $Y$ under $\RR^{-1}$:
- $\RR^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \tuple {t, s} \in \RR^{-1} }$
If no element of $Y$ has a preimage, then $\RR^{-1} \sqbrk Y = \O$.
Preimage of Subset as Element of Inverse Image Mapping
The preimage of $Y$ under $\RR$ can be seen to be an element of the codomain of the inverse image mapping $\RR^\gets: \powerset T \to \powerset S$ of $\RR$:
- $\forall Y \in \powerset T: \map {\RR^\gets} Y := \set {s \in S: \exists t \in Y: \tuple {s, t} \in \RR}$
Thus:
- $\forall Y \subseteq T: \RR^{-1} \sqbrk Y = \map {\RR^\gets} Y$
and so the preimage of $Y$ under $\RR$ is also seen referred to as the inverse image of $Y$ under $\RR$.
Both approaches to this concept are used in $\mathsf{Pr} \infty \mathsf{fWiki}$.
Also known as
The preimage of $Y$ is also known as the inverse image of $Y$.
The term preimage set is also seen.
Also see
- Preimage of Subset under Relation equals Union of Preimages of Elements
- Definition:Inverse Image Mapping of Relation
Special Cases
Generalizations
Related Concepts
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.20$