Easy set partition problem

Question

So I have this problem :

Let A be a non empty set an P1 and P2 be two random partitions of the set A. Prove that the set is also a partition of A.

I know that this is probably very easy to most of you but discrete mathematics isn't my strong side so basically i have no idea. All help is well welcomed.

Answer 1

Really all you need to do is to verify that the three axioms of a partition holds for $\cal S$.

1. $\varnothing\notin\cal S$. That's trivial by the definition.
2. $A=\bigcup\cal S$, or for every $a\in A$ there is some $Z\in\cal S$ such that $a\in Z$. But recall that $P_1$ and $P_2$ were partitions. You can find $X\in P_1, Y\in P_2$ which include $a$, and so $Z$ must be...?
3. Every two distinct sets in $\cal S$ are disjoint. This is not hard either, $(X_1\cap Y_1)\cap(X_2\cap Y_2)=(X_1\cap X_2)\cap(Y_1\cap Y_2)=\ldots$