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:


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

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 $f$:


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

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

and so the preimage of $Y$ under $f$ is also seen referred to as the inverse image of $Y$ under $f$.

Both approaches to this concept are used in.

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

Also see

 * Definition:Inverse Image Mapping of Mapping


 * Definition:Domain of Mapping
 * Definition:Codomain of Mapping
 * Definition:Range of Relation


 * Preimage of Subset under Mapping equals Union of Preimages of Elements

Generalizations

 * Definition:Preimage of Mapping


 * Definition:Preimage of Relation
 * Definition:Preimage of Subset under Relation

Related Concepts

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