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

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

## Definition

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

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

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

- $\mathcal R \left [{X}\right] := \set {t \in T: \exists s \in X: \left({s, t}\right) \in \mathcal R}$

That is:

- $\mathcal R \left [{X}\right] := \mathcal R^\to \left ({X}\right)$

where $\mathcal R^\to$ denotes the mapping induced on the power set of $S$ by $\mathcal R$.

## Also known as

The **image of $X$ under $\mathcal R$** is also seen referred to as the **direct image of $X$ under $\mathcal R$**.

## Also denoted as

As well as using the notation $\Img {\mathcal R}$ to denote the image set of a relation, the symbol $\operatorname {Img}$ can also be used as follows:

For $X \subseteq S$:

- $\operatorname {Img}_\mathcal R \paren X := \mathcal R \left [{X}\right]$

but this notation is rarely seen.

## 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

## 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): $\text{I}$: Problem $\text{AA}$ - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.11$: Relations: Definition $11.3$