The following exercise shows that Dynkin systems and \(\sigma\)-algebras are, in general, different. Let \(X=\{1,2,3, \ldots, 4 k-1,4 k\}\) for some \(k \in \mathbb{N}\). Then \(\mathscr{D}=\{A \subset X: \# A\) is even \(\}\) is a Dynkin system, but not a \(\sigma\)-algebra.