Definition:Image (Relation Theory)/Mapping/Subset

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

Let $X \subseteq S$ be a subset of $S$.

Then the image of $X$ (under $f$) is defined and denoted as:
 * $f \sqbrk X := \set {t \in T: \exists s \in X: \map f s = t}$

Image of Subset as Element of Direct Image Mapping
The image of $X$ under $f$ can be seen to be an element of the codomain of the direct image mapping $f^\to: \powerset S \to \powerset T$ of $f$:


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

Thus:
 * $\forall X \subseteq S: f \sqbrk X = \map {f^\to} X$

and so the image of $X$ under $f$ is also seen referred to as the direct image of $X$ under $f$.

Both approaches to this concept are used in.

Also known as
The term image set is often seen for image.

The modifier by $f$ can also be used for under $f$.

Thus, for example, the image set of $X$ by $f$ means the same as the image of $X$ under $f$.

Also see

 * Image of Singleton under Mapping
 * Image of Domain of Mapping is Image Set
 * Image of Subset under Mapping equals Union of Images of Elements


 * Definition:Direct Image Mapping of Mapping
 * Definition:Covariant Power Set Functor

Generalizations

 * Definition:Image of Mapping


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

Related Concepts

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