Union of Subsets is Subset/Proof 1

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: $\text{(g)}$