Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 25J / c

${\mathcal{E}}_1$ is a Dynkin system but not a pi-system. The whole ten-point set has even cardinality, complements preserve parity, and disjoint unions of even sets are even. Two even sets can intersect in a singleton, for example $\{1,2\}$ and $\{2,3\}$.

${\mathcal{E}}_2$ is neither. The infinite set $\mathbb{N}\setminus \{1\}$ belongs to it, but its singleton complement does not, so it is not a Dynkin system. The two infinite sets $\{1\}\cup \{2,4,6,\dots \}$ and $\{1,3,5,\dots \}$ intersect in the excluded singleton, so it is not a pi-system.

${\mathcal{E}}_3$ is a pi-system but not a Dynkin system. Intersections of bounded open intervals are bounded open intervals or empty. However, the family does not contain the whole underlying set $\mathbb{R}$. None of the three families is a sigma-algebra.