Definition:Image (Relation Theory)

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 $$\mathcal{R}$$ of a relation $$\mathcal{R} \subseteq S \times T$$ is the set:

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

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:

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

That is, $$\mathcal{R} \left ({s}\right)$$ is the set of all elements of $$\mathrm {Dom} \left({\mathcal{R}}\right)$$ 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:

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

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

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

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 $$\mathrm {Im} \left ({A}\right)$$ 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.

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 $$\left\{ {t}\right\}$$".