Properties of Algebras of Sets

Theorem
If $$\mathfrak{A} \ $$ is an algebra of sets, then the following hold:


 * 1) The intersection of two sets in $$\mathfrak{A} \ $$ is in $$\mathfrak{A} \ $$.
 * 2) The difference of two sets in $$\mathfrak{A} \ $$ is in $$\mathfrak{A} \ $$.

Proof
Let $$A, B \in \mathfrak{A}$$.

By the definition of algebra of sets, we have that:


 * 1) $$A \cup B \in \mathfrak{A} \ $$;
 * 2) $$\mathcal{C} \left({A}\right) \in \mathfrak{A} \ $$.

Thus:

$$ $$ $$ $$

and so we have that the intersection of two sets in $$\mathfrak{A} \ $$ is in $$\mathfrak{A} \ $$.

Next:

$$ $$ $$

and so we have that the difference of two sets in $$\mathfrak{A} \ $$ is in $$\mathfrak{A} \ $$.