Definition:Image (Relation Theory)

Definition
The definition of a relation given here 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.

Image of a Relation
The image (or image set) of a relation $\mathcal R \subseteq S \times T$ is the set:


 * $\operatorname{Im} \left ({\mathcal R}\right) = \mathcal R \left ({S}\right) = \left\{ {t \in T: \exists s \in S: \left({s, t}\right) \in \mathcal R}\right\}$

Image of an Element
For any relation $\mathcal R \subseteq S \times T$, the image of $s \in S$ by $\mathcal R$ is defined as:


 * $\operatorname{Im} \left ({s}\right) = \mathcal R \left ({s}\right) = \left\{ {t \in T: \left({s, t}\right) \in \mathcal R}\right\}$

That is, $\mathcal R \left ({s}\right)$ is the set of all elements of the codomain of $\mathcal R$ related to $s$ by $\mathcal R$.

Image of a Subset
For any relation $\mathcal R \subseteq S \times T$, the image of $A \subseteq S$ by $\mathcal R$ is:


 * $\operatorname{Im} \left ({A}\right) = \mathcal R \left ({A}\right) = \left\{ {t \in T: \exists s \in A: \left({s, t}\right) \in \mathcal R}\right\}$

If $A = \operatorname{Dom} \left({\mathcal R}\right)$, we have:


 * $\operatorname{Im} \left ({\operatorname{Dom} \left({\mathcal R}\right)}\right) = \mathcal R \left ({\operatorname{Dom} \left({\mathcal R}\right)}\right) = \operatorname{Im} \left ({\mathcal R}\right)$

It is also clear that $\forall s \in S: \mathcal R \left ({s}\right) = \mathcal R \left ({\left\{{s}\right\}}\right)$.

While the use of $\operatorname{Im} \left ({A}\right)$ etc. can be useful, it is arguably preferable in some situations to use $\mathcal R \left ({A}\right)$, as this makes it more apparent to exactly what relation the image refers. This is the terminology which we are planning to use from here on in.

Some authors use $\mathcal R^\to \left ({A}\right)$ for what we have here as $\mathcal R \left ({A}\right)$.

If Image is a Singleton
If $\mathcal R \left ({s}\right)$ for some $s \in S$ (or $\mathcal R \left ({S_1}\right)$ for some $S_1 \subseteq S$) has only one element $t \in T$, then we can write:


 * $\mathcal R \left ({s}\right) = t$

instead of:


 * $\mathcal R \left ({s}\right) = \left\{{t}\right\}$

Note: The two notations $s \mathcal R t$ and $\mathcal R \left ({s}\right) = t$ do not mean the same thing.

The first means: "$s$ is related to $t$ by $\mathcal R$" (which does not exclude the possibility of there being other elements of $T$ to which $s$ relates).

The second means "The complete set of elements of $T$ to which $s$ relates consists of $\left\{ {t}\right\}$".

Also see

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


 * Domain
 * Codomain
 * Range


 * Preimage (also known as inverse image)