Set is Subset of Intersection of Supersets/Proof 1

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

Let $S \subseteq T_1 \land S \subseteq T_2$.

Then:

\(\ds x \in S\)

\(\leadsto\)

\(\ds x \in T_1 \land x \in T_2\)

Definition of Subset

\(\ds \)

\(\leadsto\)

\(\ds x \in T_1 \cap T_2\)

Definition of Set Intersection

\(\ds \)

\(\leadsto\)

\(\ds S \subseteq T_1 \cap T_2\)

Definition of Subset

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.3$. Intersection: Example $13$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 7.2 \ \text{(i)}$: Unions and Intersections