Definition:Image (Relation Theory)/Relation
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Definition
Let $\RR \subseteq S \times T$ be a relation.
Image of a Relation
The image of $\RR$ is defined as:
- $\Img \RR := \RR \sqbrk S = \set {t \in T: \exists s \in S: \tuple {s, t} \in \RR}$
Image of an Element
Let $s \in S$.
The image of $s$ by (or under) $\RR$ is defined as:
- $\map \RR s := \set {t \in T: \tuple {s, t} \in \RR}$
That is, $\map \RR s$ is the set of all elements of the codomain of $\RR$ related to $s$ by $\RR$.
Image of a Subset
Let $X \subseteq S$ be a subset of $S$.
Then the image set (of $X$ by $\RR$) is defined as:
- $\RR \sqbrk X := \set {t \in T: \exists s \in X: \tuple {s, t} \in \RR}$
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 $\RR$ directly to a subset $A$, those sources prefer to define the direct image mapping of $\RR$ as a separate concept in its own right.
Also see
- Definition:Preimage of Relation (also known as Definition:Inverse Image)
Special cases
- Definition:Image of Mapping, in which the context of an image is usually encountered.