Symbols:I

Image

$\operatorname{Im}$

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 $\forall s \in S: \mathcal R \left ({s}\right) = \mathcal R \left ({\left\{{s}\right\}}\right)$.

Mapping

Image of a Mapping

The image (or image set) of a mapping $f: S \to T$ is the set:

$\operatorname{Im} \left ({f}\right) = f \left ({S}\right) = \left\{ {t \in T: \exists s \in S: f \left({s}\right) = t}\right\}$

Image of an Element

Let $s \in S$.

The image of $s$ by (or under) $f$ is defined as:

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

That is, $f \left ({s}\right)$ is the element of the codomain of $f$ related to $s$ by $f$.

Image of a Subset

Let $X \subseteq S$.

Then the image (or image set) of $X$ (by $f$) is defined as:

$\operatorname {Im} \left ({X}\right) := \left\{ {t \in T: \exists s \in X: f \left({s}\right) = t}\right\}$

If $X = \operatorname{Dom} \left({f}\right)$, we have:

$\operatorname{Im} \left ({\operatorname{Dom} \left({f}\right)}\right) = \operatorname{Im} \left ({f}\right)$

where $\operatorname{Im} \left ({f}\right)$ is the image (set) of $f$.

The $\LaTeX$ code for $\operatorname{Im} \left ({\mathcal R}\right)$ is \operatorname{Im} \left ({\mathcal R}\right).