# Properties of Algebras of Sets

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## 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 Intersectionâ€Ž

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.