# Set Union/Examples/Subset of Union

Jump to navigation
Jump to search

## 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

\(\displaystyle w\) | \(\in\) | \(\displaystyle W\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle w\) | \(\in\) | \(\displaystyle U\) | by hypothesis | |||||||||

\(\, \displaystyle \lor \, \) | \(\displaystyle w\) | \(\in\) | \(\displaystyle V\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle w\) | \(\in\) | \(\displaystyle U \cup V\) | Definition of Set Union | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle W\) | \(\subseteq\) | \(\displaystyle U \cup V\) | Definition of Subset |

$\blacksquare$

## Sources

- 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.1$: Sets and Subsets: Problem Set $\text{A}.1$: $2$