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Proof

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

Context

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

Hypothesis

• Let $x\in A$.

Implications

• 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, $x\in A\cup B.$

Conclusion

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

Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016