$X = A \cup B$ and $\mathfrak{A}(\mathfrak{B}$, respectively) is a $\sigma$-algebra of subsets of $A$ ($B$, respectively). Prove that $$\mathfrak{D} = \{E\cup F:E\in \mathfrak{A}, F \in \mathfrak{B}\}$$ is a $\sigma$ algebra of subsets from $X$.
It's not difficult to show that $X$, $\emptyset \in \mathfrak{D}$. I find it more challenging, however, to show that any countable union of $C_n \in \mathfrak{D}$ lies in $\mathfrak{D}$.
So, for each $n$ there exist $E_n$ and $F_n$ such that $E_n \in \mathfrak{A}, F_n \in \mathfrak{B}$ and $C_n = E_n\cup F_n$
So $$\bigcup C_n = \underbrace{(\bigcup E_n)}_{\in\mathfrak{A} }\cup \underbrace{(\bigcup F_n)}_{\in\mathfrak{B} }\in \mathfrak{D } $$