Q:

A trivial example of when the union of two sigma algebras is not a sigma algebra

Consider the following statement:
If $\sum_{1}$, $\sum_{2}$ are $\sigma-$ algebras,
then $\Sigma_{1}\cup\Sigma_{2}$ is in general not a $\sigma-$algbra.
In order to come up with an illustrative example of when this union is not a sigma algebra, I consider two very trivial sigma algebras:
Consider $\sum_{1}=\{\emptyset,S_{1}\}$ and $\Sigma_{2}=\{\emptyset,S_{2}\}$
then $\Sigma_{1}\cup\Sigma_{2}=\{\emptyset,S_{1},S_{2}\}$ .The union is by no means a sigma algebra- it does not contain the modified new sample space (presumably $S_{1}\cup S_{2}$)  nor the fact that given that $S_{1}$ and $S_{2}$  are contained in the union, their union is not. Is this reasoning correct?

A:

$\Sigma_1 = \{\emptyset, \{a\}, \{b, c\}, \{a, b, c\}\}$ and $\Sigma_2 = \{\emptyset, \{a, b\}, \{c\}, \{a, b, c\}\}$ is a trivial example of two $\sigma$-algebras on the same set whose union is not a $\sigma$-algebra: the union contains $\{a, b\}$ and $\{b, c\}$ but not their intersection.

