# Image of Union under Relation/Family of Sets

## Theorem

Let $S$ and $T$ be sets.

Let $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ be a family of subsets of $S$.

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

Then:

$\displaystyle \mathcal R \left[{\bigcup_{i \mathop \in I} S_i}\right] = \bigcup_{i \mathop \in I} \mathcal R \left[{S_i}\right]$

where $\displaystyle \bigcup_{i \mathop \in I} S_i$ denotes the union of $\left\langle{S_i}\right\rangle_{i \in I}$.

## Proof

 $\displaystyle t$ $\in$ $\displaystyle \mathcal R \left[{\bigcup_{i \mathop \in I} S_i}\right]$ $\displaystyle \iff \ \$ $\displaystyle \exists s \in \bigcup_{i \mathop \in I} S_i: t$ $\in$ $\displaystyle \mathcal R \left[{s}\right]$ Image of Subset under Relation equals Union of Images of Elements $\displaystyle \iff \ \$ $\displaystyle \exists i \in I: \exists s \in S_i: t$ $\in$ $\displaystyle \mathcal R \left[{s}\right]$ Definition of Set Union $\displaystyle \iff \ \$ $\displaystyle \exists i \in I: t$ $\in$ $\displaystyle \mathcal R \left[{S_i}\right]$ Definition of Image of Subset $\displaystyle \iff \ \$ $\displaystyle t$ $\in$ $\displaystyle \bigcup_{i \mathop \in I} \mathcal R \left[{S_i}\right]$ Definition of Set Union

$\blacksquare$