Set Union/Examples/Subset of Union

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Example of Set Union

Let $U, V, W$ be non-empty sets.

Let $W$ be such that for all $w \in W$, either:

$w \in U$

or:

$w \in V$

Then:

$W \subseteq U \cup V$


Proof

\(\displaystyle w\) \(\in\) \(\displaystyle W\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle w\) \(\in\) \(\displaystyle U\) $\quad$ by hypothesis $\quad$
\(\, \displaystyle \lor \, \) \(\displaystyle w\) \(\in\) \(\displaystyle V\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle w\) \(\in\) \(\displaystyle U \cup V\) $\quad$ Definition of Set Union $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle W\) \(\subseteq\) \(\displaystyle U \cup V\) $\quad$ Definition of Subset $\quad$

$\blacksquare$


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