Image of Union under Relation/Family of Sets

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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

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

$\blacksquare$


Sources