Definition:Preimage/Mapping/Subset

Definition
Let $f: S \to T$ be a mapping.

Let $f^{-1} \subseteq T \times S$ be the inverse of $f$, defined as:


 * $f^{-1} = \set {\tuple {t, s}: \map f s = t}$

Let $Y \subseteq T$.

The preimage of $Y$ under $f$ is defined as:


 * $f^{-1} \sqbrk Y := \set {s \in S: \exists y \in Y: \map f s = y}$

That is, the preimage of $Y$ under $f$ is the image of $Y$ under $f^{-1}$, where $f^{-1}$ can be considered as a relation.

If no element of $Y$ has a preimage, then $f^{-1} \sqbrk Y = \O$.

Preimage of Subset as Element of Inverse Image Mapping
The preimage of $Y$ under $f$ can be seen to be an element of the codomain of the inverse image mapping $f^\gets: \powerset T \to \powerset S$ of $f$:


 * $\forall Y \in T: \map {f^\gets} Y := \set {s \in S: \exists y \in Y: \map f s = y}$

Both approaches to this concept are used in

Also known as
Some sources use counter image or inverse image instead of preimage.

Also denoted as
When the language of induced mappings is used, then $\map {f^\gets} Y$ is seen for $f^{-1} \sqbrk Y$.

Also see

 * Definition:Image of Subset under Mapping
 * Definition:Preimage Mapping
 * Definition:Domain of Mapping
 * Definition:Codomain of Mapping
 * Definition:Range
 * Preimage of Subset under Mapping equals Union of Preimages of Elements

Generalization

 * Definition:Preimage of Relation