Union of Subsets is Subset/Proof 1
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Theorem
Let $S_1$, $S_2$, and $T$ be sets.
Let $S_1$ and $S_2$ both be subsets of $T$.
Then:
- $S_1 \cup S_2 \subseteq T$
That is:
- $\paren {S_1 \subseteq T} \land \paren {S_2 \subseteq T} \implies \paren {S_1 \cup S_2} \subseteq T$
Proof
Let:
- $\paren {S_1 \subseteq T} \land \paren {S_2 \subseteq T}$
Then:
\(\ds S_1 \cup S_2\) | \(\subseteq\) | \(\ds T \cup T\) | Set Union Preserves Subsets | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds S_1 \cup S_2\) | \(\subseteq\) | \(\ds T\) | Set Union is Idempotent |
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: $\text{(g)}$