# Set Union/Examples/Subset of Union

## 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 $\quad$

$\blacksquare$