# Image of Subset under Relation equals Union of Images of Elements

## Theorem

Let $S$ and $T$ be sets.

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

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

Then:

$\displaystyle \mathcal R \left[{X}\right] = \bigcup_{x \mathop \in X} \mathcal R \left({x}\right)$

where:

$\mathcal R \left[{X}\right]$ is the image of the subset $X$ under $\mathcal R$
$\mathcal R \left({x}\right)$ is the image of the element $x$ under $\mathcal R$.

## Proof

By definition:

$\mathcal R \left[{X}\right] = \left\{ {y \in T: \exists x \in X: \left({x, y}\right) \in \mathcal R}\right\}$
$\mathcal R \left({x}\right) = \left\{ {y \in T: \left({x, y}\right) \in \mathcal R}\right\}$

First:

 $\displaystyle y$ $\in$ $\displaystyle \mathcal R \left[{X}\right]$ $\displaystyle \implies \ \$ $\displaystyle \exists x \in X: \left({x, y}\right)$ $\in$ $\displaystyle \mathcal R$ Definition of $\mathcal R \left[{X}\right]$ $\displaystyle \implies \ \$ $\displaystyle \left({x, y}\right)$ $\in$ $\displaystyle \bigcup_{x \mathop \in X} \left\{ {\left({x, y}\right) \in \mathcal R}\right\}$ Definition of Set Union $\displaystyle \implies \ \$ $\displaystyle y$ $\in$ $\displaystyle \bigcup_{x \mathop \in X} \left\{ {y \in T: \left({x, y}\right) \in \mathcal R}\right\}$ Definition of Relation $\displaystyle \implies \ \$ $\displaystyle y$ $\in$ $\displaystyle \bigcup_{x \mathop \in X} \mathcal R \left({x}\right)$ Definition of $\mathcal R \left({x}\right)$ $\displaystyle \implies \ \$ $\displaystyle \mathcal R \left[{X}\right]$ $\subseteq$ $\displaystyle \bigcup_{x \mathop \in X} \mathcal R \left({x}\right)$ Definition of Subset

Then:

 $\displaystyle y$ $\in$ $\displaystyle \bigcup_{x \mathop \in X} \mathcal R \left({x}\right)$ $\displaystyle \implies \ \$ $\displaystyle y$ $\in$ $\displaystyle \bigcup_{x \mathop \in X} \left\{ {y \in T: \left({x, y}\right) \in \mathcal R}\right\}$ Definition of $\mathcal R \left({x}\right)$ $\displaystyle \implies \ \$ $\displaystyle \exists x \in X: \left({x, y}\right)$ $\in$ $\displaystyle \mathcal R$ Definition of Set Union $\displaystyle \implies \ \$ $\displaystyle y$ $\in$ $\displaystyle \left\{ {y \in T: \exists x \in X: \left({x, y}\right) \in \mathcal R}\right\}$ Definition of Relation $\displaystyle \implies \ \$ $\displaystyle y$ $\in$ $\displaystyle \mathcal R \left[{X}\right]$ Definition of $\mathcal R \left[{X}\right]$ $\displaystyle \implies \ \$ $\displaystyle \bigcup_{x \mathop \in X} \mathcal R \left({x}\right)$ $\subseteq$ $\displaystyle \mathcal R \left[{X}\right]$ Definition of Subset

So:

$\displaystyle \bigcup_{x \mathop \in X} \mathcal R \left({x}\right) \subseteq \mathcal R \left[{X}\right]$

and:

$\displaystyle \mathcal R \left[{X}\right] \subseteq \bigcup_{x \mathop \in X} \mathcal R \left({x}\right)$

The result follows by definition of set equality.

$\blacksquare$