# Deducing additional set theoretic properties from definition of dynkin system

I am trying to prove certain set theoretic properties assuming dynkin system. Definition of Dynkin system - Let $$X$$ be a non empty set and let D be a collection of subsets of $$X$$, Then $$D$$ is a dynkin system if -

1. $$X \in D$$ .
2. If $$A$$ and $$B$$ belongs to $$D$$ and $$A \subseteq B$$, then $$B \setminus A$$ belongs to $$D$$.
3. If $$A_1, A_2, \dots$$ is a sequence of subsets such that $$A_n \subseteq A_{n+1}$$ for all $$n \ge 1$$ , then $$\bigcup\limits_{n =1}^{\infty}A_n \in D$$.

In following properties to be proved properties I don't know how they are true or please give some Counterexample. I am only interested how to prove them if intersection of two arbitrary sets assumed is non empty as I have already proved for disjoint sets. Properties are -

1. Assume sets $$A$$ and $$B$$ belong to $$D$$ and they have non empty intersection, does $$A \cap B$$ always belong to $$D$$.
2. Assume sets $$A$$ and $$B$$ belong to $$D$$ and they have non empty intersection , does $$A \cup B$$ always belong to $$D$$ .
• @SimoneRamello hope question is fine now. Can you please give some hints. I am really struct.
– user775699
Sep 2, 2019 at 8:01

Let $$A,B\subset X$$ such that the sets $$A\cap B$$, $$A\cap B^{\complement}$$, $$A^{\complement}\cap B$$ and $$A^{\complement}\cap B^{\complement}$$ are not empty.

Let $$\mathcal D:=\{\varnothing, A,A^{\complement},B, B^{\complement},X\}$$.

Then $$\mathcal D$$ is a Dynkin-system but this with $$A\cap B\notin\mathcal D$$ and $$A\cup B\notin\mathcal D$$.

Observe that for $$D_1,D_2\in\mathcal D$$ we have $$D_1\subseteq D_2$$ if and only if one of the following conditions is satisfied:

• $$D_1=D_2$$
• $$D_1=\varnothing$$
• $$D_2=X$$

This makes it easy to verify that $$\mathcal D$$ is indeed a Dynkin-system.