Determine if the collection of set formed by element by element union,intersection and set difference of two sigma algebra is a sigma algebra

Question

Let $F_1 * F_2 = \{A : A=A_1 \cup A_2, \:A_1 \in F_1 \land A_2 \in F_2\}$. If $F_1$ and $F_2$ are $\sigma$-algebra (fields), is $F_1 * F_2$ a $\sigma$-algebra?

Similarly what about element-by-element intersection and set differences?

Answer 1

Hint: What happens when $F_1$ and $F_2$ are the $\sigma$-algebras on the set $X=\{1,2,3,4\}$ given by $F_1=\{\emptyset, X, \{1,2\},\{3,4\}\}$ and $F_2=\{\emptyset,X,\{1,3\},\{2,4\}\}$?