# Image of Union under Relation

## Theorem

Let $S$ and $T$ be sets.

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

Let $S_1$ and $S_2$ be subsets of $S$.

Then:

$\mathcal R \sqbrk {S_1 \cup S_2} = \mathcal R \sqbrk {S_1} \cup \mathcal R \sqbrk {S_2}$

That is, the image of the union of subsets of $S$ is equal to the union of their images.

### General Result

Let $S$ and $T$ be sets.

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

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S \subseteq \mathcal P \left({S}\right)$.

Then:

$\displaystyle \mathcal R \left[{\bigcup \mathbb S}\right] = \bigcup_{X \mathop \in \mathbb S} \mathcal R \left[{X}\right]$

### Family of Sets

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

 $\displaystyle t$ $\in$ $\displaystyle \mathcal R \sqbrk {S_1 \cup S_2}$ $\displaystyle \iff \ \$ $\displaystyle \exists s \in S_1 \cup S_2: t$ $\in$ $\displaystyle \mathcal R \sqbrk s$ Definition of Image of Subset under Relation $\displaystyle \iff \ \$ $\displaystyle \exists s: s \in S_1 \lor s \in S_2: t$ $\in$ $\displaystyle \mathcal R \sqbrk s$ Definition of Set Union $\displaystyle \iff \ \$ $\displaystyle t$ $\in$ $\displaystyle \mathcal R \sqbrk {S_1} \lor t \in \mathcal R \sqbrk {S_2}$ Definition of Image of Subset under Relation $\displaystyle \iff \ \$ $\displaystyle t$ $\in$ $\displaystyle \mathcal R \sqbrk {S_1} \cup \mathcal R \sqbrk {S_2}$ Definition of Set Union

$\blacksquare$