Set is Subset of Intersection of Supersets/Proof 2

Theorem

Let $S$, $T_1$ and $T_2$ be sets.

Let $S$ be a subset of both $T_1$ and $T_2$.

Then:

$S \subseteq T_1 \cap T_2$

That is:

$\paren {S \subseteq T_1} \land \paren {S \subseteq T_2} \implies S \subseteq \paren {T_1 \cap T_2}$

Proof

\(\ds S\)

\(\subseteq\)

\(\ds T_1\)

\(\, \ds \land \, \)

\(\ds S\)

\(\subseteq\)

\(\ds T_2\)

\(\ds \leadsto \ \ \)

\(\ds S \cap S\)

\(\subseteq\)

\(\ds S \cap T_2\)

Set Intersection Preserves Subsets

\(\ds \leadsto \ \ \)

\(\ds S\)

\(\subseteq\)

\(\ds T_1 \cap T_2\)

Set Intersection is Idempotent

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: $\text{(g)}$