Let \(M\) be a set and let \({\mathcal {R}}\) be a system of its subsets \({\mathcal {R}}\) is called a ring of sets, if is is closed under set-theoretic differences and set-theoretic unions, i.e. if
\[S,T\in {\mathcal {R}}\Longrightarrow \cases{S\setminus T\in {\mathcal {R}},\\S\cup T\in {\mathcal {R}}.}\]
Please note that in particular \(\emptyset \in {\mathcal {R}}\), if we take \(S=T\).