Properties of Algebras of Sets

Theorem

Let $S$ be a set.

Let $\RR$ be an algebra of sets on $S$.

Then the following hold:

Algebra of Sets is Closed under Intersection‎

Let $S$ be a set.

Let $\RR$ be an algebra of sets on $S$.

Then:

$\forall A, B \in S: A \cap B \in \RR$

Algebra of Sets is Closed under Set Difference

Let $S$ be a set.

Let $\RR$ be an algebra of sets on $S$.

Then:

$\forall A, B \in S: A \setminus B \in \RR$

Algebra of Sets contains Underlying Set

Let $S$ be a set.

Let $\RR$ be an algebra of sets on $S$.

Then:

$S \in \RR$

Algebra of Sets contains Empty Set

Let $S$ be a set.

Let $\RR$ be an algebra of sets on $S$.

Then:

$\O \in \RR$

where $\O$ denotes the empty set.