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.

Then the image of $X$ (under $f$) is defined as:
 * $f \left [{X}\right] := \set {t \in T: \exists s \in X: f \paren s = t}$

That is:
 * $f \left [{X}\right] := f^\to \paren X$

where $f^\to$ denotes the mapping induced on the power set of $S$ by $f$.

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 $f$.

Some authors stipulate the name further by calling $f \left [{X}\right]$ the direct image of $X$ (by $f$).

This is done to emphasize the distinction between this and the concept of the inverse image.

Also see

 * Definition:Image of Subset under Relation


 * 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
 * Definition:Covariant Power Set Functor