Image of Union

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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:

$\mathcal R \left[{S_1 \cup S_2}\right] = \mathcal R \left[{S_1}\right] \cup \mathcal R \left[{S_2}\right]$


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 $\left\langle{S_i}\right\rangle_{i \in I}$ 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

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle t\) \(\in\) \(\displaystyle \) \(\displaystyle \mathcal R \left[{S_1 \cup S_2}\right]\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \iff\) \(\displaystyle \) \(\displaystyle \exists s \in S_1 \cup S_2: t\) \(\in\) \(\displaystyle \) \(\displaystyle \mathcal R \left[{s}\right]\) \(\displaystyle \) \(\displaystyle \)          Definition of image of subset          
\(\displaystyle \) \(\displaystyle \iff\) \(\displaystyle \) \(\displaystyle \exists s: s \in S_1 \lor s \in S_2: t\) \(\in\) \(\displaystyle \) \(\displaystyle \mathcal R \left[{s}\right]\) \(\displaystyle \) \(\displaystyle \)          Definition of union          
\(\displaystyle \) \(\displaystyle \iff\) \(\displaystyle \) \(\displaystyle t\) \(\in\) \(\displaystyle \) \(\displaystyle \mathcal R \left[{S_1}\right] \lor t \in \mathcal R \left[{S_2}\right]\) \(\displaystyle \) \(\displaystyle \)          Definition of image of subset          
\(\displaystyle \) \(\displaystyle \iff\) \(\displaystyle \) \(\displaystyle t\) \(\in\) \(\displaystyle \) \(\displaystyle \mathcal R \left[{S_1}\right] \cup \mathcal R \left[{S_2}\right]\) \(\displaystyle \) \(\displaystyle \)          Definition of union          

$\blacksquare$


Also see


Sources