TITLE: Union of two $\sigma$-algebras is not $\sigma$-algebra
QUESTION [14 upvotes]: Here is another very basic analysis problem but that puzzles me:

Find an example of set $X$ and its two $\sigma$-algebras $\mathscr A_1$ and $\mathscr A_2$, such that $\mathscr A_1 \cup \mathscr A_2$ is not $\sigma$-algebra.

To me at least, this question looks counter-intuitive since the union of two sets gives the resulting set larger number of elements, thus won't affect its $\sigma$-algebra status.
Please help and thank you for your time and effort.

REPLY [2 votes]: $\{\emptyset,[0,3/4),[3/4,1),[0,1)\}$ and $\{\emptyset,[0,1/2),[1/2,1),[0,1)\}$ are $\sigma-$algebras but their union $\{\emptyset,[0,3/4),[3/4,1),[0,1/2),[1/2,1),[0,1)\}$ is not. $[0,3/4)\ minus \ [1/2,1)=[1/2,3/4)$ is not in the union.