Review Note

Remark 1.1.3
a) Rings are stable only with respect to {{c1::relative complement, whereas algebras are stable under complement in \(X\)}}

b) A ring contains with each two its sets not only their union but also {{c2::their intersection. This is because \(A \cap B = A \setminus(A \setminus B)\)}}

c) A ring \(A \subset \mathcal P(X)\) is an algebra if and only if it has the following properties:
{{c3::1) \(X \in \mathcal A\)
2) \(A, B \in \mathcal A \implies A \cup B \in \mathcal A\)
3) \(A \in \mathcal A \implies A^C \in \mathcal A\)}}

d) A ring is also stable under {{c4::finite unions and intersections}}