Definition:Image (Set Theory)/Relation

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Definition

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


Image of a Relation

The image of $\mathcal R$ is the set:

$\Img {\mathcal R} := \mathcal R \sqbrk S = \set {t \in T: \exists s \in S: \tuple {s, t} \in \mathcal R}$


Image of an Element

Let $s \in S$.

The image of $s$ by (or under) $\mathcal R$ is defined as:

$\map {\mathcal R} s := \set {t \in T: \tuple {s, t} \in \mathcal R}$

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


Image of a Subset

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


Then the image set (of $X$ by $\mathcal R$) is defined as:

$\mathcal R \sqbrk X := \set {t \in T: \exists s \in X: \tuple {s, t} \in \mathcal R}$


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 direct image mapping of $\mathcal R$ as a separate concept in its own right.


Also see


Special cases