My professor proved the following theorem:
Let $\mathcal{E}$ be an elementary family of subsets of (non-empty) set $\Bbb{X}$. Then, the collection of finite disjoint unions of members of $\mathcal{E}$,
$\mathcal{A}$ = $\{\displaystyle \sqcup_{i=1}^{n} E_k\ |\ n \in \Bbb{N}, E_1, ..., E_n \in \mathcal{E} \}$ is an algebra.
He proved it by showing that $\mathcal{A}$ is closed under pairwise intersection, closed under disjoint pairwise union and closed under complements.
My question is why did he prove it like that? Isn't it enough to prove that $\mathcal{A}$ is closed under finite union and complementation.
We have to show that $\mathcal{A}$ is closed under finite union if we assume that it is closed under pairwise intersection, pairwise disjoint union and complementation. We can show it for the union of two sets (the general case follows by induction). So let us consider two sets $X,Y \in \mathcal{A}$. Denote $W:=X\cap Y \in \mathcal{A}$. Then
$$ X \cup Y = (X \cap W^c) \sqcup (Y \cap W^c) \in \mathcal{A}.$$