Definition:Image (Set Theory)/Relation/Subset

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.