Algebra of Sets is Closed under Intersection/Proof 1
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Theorem
Let $S$ be a set.
Let $\RR$ be an algebra of sets on $S$.
Then:
- $\forall A, B \in S: A \cap B \in \RR$
Proof
By definition $2$ of Algebra of Sets:
- $\RR$ is a ring of sets with a unit.
By definition $1$ of Ring of Sets:
\((\text {RS} 2_1)\) | $:$ | Closure under Intersection: | \(\ds \forall A, B \in \RR:\) | \(\ds A \cap B \in \RR \) |
$\blacksquare$