# Definition:Image (Relation Theory)/Relation/Subset

< Definition:Image (Relation Theory) | Relation(Redirected from Definition:Image of Subset under Relation)

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## Definition

Let $\RR \subseteq S \times T$ be a relation.

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

Then the **image set (of $X$ by $\RR$)** is defined as:

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

### Image of Subset as Element of Direct Image Mapping

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The **image of $X$ by $\RR$** can be seen to be an element of the codomain of the direct image mapping $\RR^\to: \powerset S \to \powerset T$ of $\RR$:

- $\forall X \in \powerset S: \map {\RR^\to} X := \set {t \in T: \exists s \in X: \tuple {s, t} \in \RR}$

Thus:

- $\forall X \subseteq S: \RR \sqbrk X = \map {\RR^\to} X$

and so the **image of $X$ under $\RR$** is also seen referred to as the **direct image of $X$ under $\RR$**.

Both approaches to this concept are used in $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also see

- Image of Subset under Relation equals Union of Images of Elements
- Image of Domain of Relation is Image Set
- Image of Singleton under Relation

### Special Cases

### Generalizations

### Related Concepts

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Relations - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Problem $\text{AA}$: Relations - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.11$: Relations: Definition $11.3$