Definition:Image
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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
Let $\mathcal R \subseteq S \times T$ be 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 $\mathcal R \subseteq S \times T$ be a relation.
Let $s \in S$.
The image of $s$ 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
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 $\forall s \in S: \mathcal R \left ({s}\right) = \mathcal R \left ({\left\{{s}\right\}}\right)$.
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.
Some authors use $\mathcal R^\to \left ({A}\right)$ for what we have here as $\mathcal R \left ({A}\right)$.
Image of a Mapping
The image (or image set) of a mapping $f: S \to T$ is the set:
- $\operatorname{Im} \left ({f}\right) = f \left ({S}\right) = \left\{ {t \in T: \exists s \in S: f \left({s}\right) = t}\right\}$
Notes
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
- Mapping, in which the context of an image is usually encountered.
- Preimage (also known as inverse image)
