Definition:Image (Relation Theory)

Definition
The definition of a relation as a subset of the Cartesian product of two sets gives a "static" sort of feel to the concept.

However, we can also consider a relation as being an operator, where you feed an element $s \in S$ (or a subset $S_1 \subseteq S$) in at one end, and you get a set of elements $T_s \subseteq T$ out of the other.

Thus we arrive at the following definition.

Also known as
Some sources refer to this as the direct image of a (usually) mapping, in order to differentiate it from an inverse image.

Rather than apply a relation $\mathcal R$ (or mapping $f$) directly to a subset $A$, those sources prefer to define the mapping induced by $\mathcal R$ or $f$ as a separate concept in its own right.

Also see

 * Definition:Mapping, in which the context of an image is usually encountered.


 * Definition:Domain (Set Theory)
 * Definition:Codomain (Set Theory)
 * Definition:Range


 * Definition:Preimage (also known as Definition:Inverse Image)