(related to Proposition: Sets are Subsets of Their Union)

- Let $A$ and $B$ be sets.
- We will prove $A\subseteq A\cup B$. The proof for $B\subseteq A\cup B$ is similar.

- Let $x\in A$.

- Since $x\in A$, by the truth table of disjunction "or" ("$\vee$") it is true that $x\in A \vee x\in B$.
- By the definition of union, $x\in A\cup B.$

- Since $x\in A\cup B$ for all $x\in A$, then by definition of subset $A\subseteq A\cup B$.∎

**Kane, Jonathan**: "Writing Proofs in Analysis", Springer, 2016