Definition:Image/Relation

Definition

Let $\mathcal R \subseteq S \times T$ be a relation.

Image of a Relation

The image (or image set) of $\mathcal R$ 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

Let $s \in S$.

The image of $s$ by (or under) $\mathcal R$ 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

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

Then the image set (of $A$ 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 $\forall s \in S: \mathcal R \left ({s}\right) = \mathcal R \left ({\left\{{s}\right\}}\right)$.

Notes

Some sources refer to this as the direct image of a relation, in order to differentiate it from an inverse image.

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

Also see

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

• Domain
• Codomain
• Range

• Preimage (also known as inverse image)

Sources

• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 4$