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

\(\ds w\)

\(\in\)

\(\ds W\)

\(\ds \leadsto \ \ \)

\(\ds w\)

\(\in\)

\(\ds U\)

by hypothesis

\(\, \ds \lor \, \)

\(\ds w\)

\(\in\)

\(\ds V\)

\(\ds \leadsto \ \ \)

\(\ds w\)

\(\in\)

\(\ds U \cup V\)

Definition of Set Union

\(\ds \leadsto \ \ \)

\(\ds W\)

\(\subseteq\)

\(\ds U \cup V\)

Definition of Subset

$\blacksquare$

Sources

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