Image of Union under Relation

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Theorem

Let $S$ and $T$ be sets.

Let $\RR \subseteq S \times T$ be a relation.

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


Then:

$\RR \sqbrk {S_1 \cup S_2} = \RR \sqbrk {S_1} \cup \RR \sqbrk {S_2}$


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 $\RR \subseteq S \times T$ be a relation.

Let $\powerset S$ be the power set of $S$.

Let $\mathbb S \subseteq \powerset S$.


Then:

$\displaystyle \RR \sqbrk {\bigcup \mathbb S} = \bigcup_{X \mathop \in \mathbb S} \RR \sqbrk X$


Family of Sets

Let $S$ and $T$ be sets.

Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$.

Let $\RR \subseteq S \times T$ be a relation.


Then:

$\displaystyle \RR \sqbrk {\bigcup_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \RR \sqbrk {S_i}$

where $\displaystyle \bigcup_{i \mathop \in I} S_i$ denotes the union of $\family {S_i}_{i \mathop \in I}$.


Proof

\(\displaystyle t\) \(\in\) \(\displaystyle \RR \sqbrk {S_1 \cup S_2}\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \exists s \in S_1 \cup S_2: t\) \(\in\) \(\displaystyle \RR \sqbrk s\) Definition of Image of Subset under Relation
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \exists s: s \in S_1 \lor s \in S_2: t\) \(\in\) \(\displaystyle \RR \sqbrk s\) Definition of Set Union
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle t\) \(\in\) \(\displaystyle \RR \sqbrk {S_1} \lor t \in \RR \sqbrk {S_2}\) Definition of Image of Subset under Relation
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle t\) \(\in\) \(\displaystyle \RR \sqbrk {S_1} \cup \RR \sqbrk {S_2}\) Definition of Set Union

$\blacksquare$


Also see


Sources