Union of elements in a sigma algebra

Question

For a sigma algebra $\sigma$, it is true that if $A \in \sigma$ and $B \in \sigma$, then $A \cup B \in \sigma$. My question is, for another $C = A' \cup B'$ where $C \in \sigma$; are $A'$ and $B'$ also part of the sigma algebra? In other words, is it true that:

$C \in \sigma \Rightarrow A' \in \sigma, B' \in \sigma$

Answer 1

False!

Let $X = \{1,2,3,4,5\}$

Then $M = \{\emptyset, \{2,3\},\{1,4,5\},X\}$ is a sigma algebra over $X$

Then $\{2,3\} \in M$ but $\{3\},\{2\} \notin M$

Answer 2

Take $B'$ to be any nonmeasurable set. Take $A'$ to be its complement (also nonmeasurable). Then $A'\cup B'=X$, your entire space which is measurable.