Union of Dynkin systems is not a Dynkin system (counterexample)

Question

If $X$ is a set and $\mathcal{D_n}\subset \mathcal{P}(X)$ are Dynkin systems, then the union $\displaystyle \bigcup_{n} \mathcal{D_n}$ is not always a Dynkin system. I need to find a counterexample for that, can anyone give me a hint?

Answer 1

$X=\{1,2,3,4,5,6\}$,

$$\mathcal D_1=\bigl\{\varnothing, X, \{1,2\}, \{3,4,5,6\}, \{1,3\}, \{2,4,5,6\}\bigr\}$$ $$\mathcal D_2=\bigl\{\varnothing, X, \{3,4\}, \{1,2,5,6\}, \{3,5\}, \{1,2,4,6\}\bigr\}$$

$\mathcal D_1\cup \mathcal D_2$ is not a Dynkin system since $\{1,2\}\cup\{3,4\}\not\in \mathcal D_1\cup \mathcal D_2$, but these are disjoint sets.