What is the smallest $\sigma$-algebra containing $[1,2]$ and $[3,4]$? What is the smallest $\sigma$-algebra containing $[1,2]$ and $[3,4]$?
 A: The smallest $\sigma$-algebra containing $[1,2]$ and $[3,4]$ is: $$\bigl\{\emptyset,[1,2],[3,4],[1,2]\cup[3,4]\bigr\}$$ I suspect, however, that you are looking for a $\sigma$-algebra on a particular superset of those two intervals. Here's how we can approach it, in general.
Suppose that $X$ is some set and $A,B$ are disjoint, non-empty, proper subsets of $X.$ In the case that $A\cup B=X,$ we have a four-element $\sigma$-algebra $\{\emptyset,A,B,X\},$ as above.
Suppose not. Any $\sigma$-algebra on $X$ containing $A$ must also contain $X\smallsetminus A,$ and we can say something similar for $B$. Any $\sigma$-algebra containing $A$ and $B$ must contain $A\cup B.$ Given the conditions, we can then show that $$\bigl\{\emptyset,A,B,A\cup B,X\smallsetminus(A\cup B),X\smallsetminus A,X\smallsetminus B,X\bigr\}$$ is the smallest $\sigma$-algebra we desire.
If we remove any of the italicized conditions above, then our smallest $\sigma$-algebra will instead have $1,2,4,$ or $16$ elements, unless $A,B$ are non-empty proper subsets of $X$ and one of them is a non-empty proper subset of the other, in which case we again have an eight-element $\sigma$-algebra, but not exactly the same one.
