I have no idea if this is correct, but this is my best estimation of what the question is asking.
A partition is such that
and i.e. is exhaustive
Then by definition of borel-algebra we have
Is this right?
(you're asked for a sigma algebra, not a Borel algebra )
So yup, the empty set and belong to (first+second axiom)
A,B,C belong to
Their complement belong to .
But... you can have a more precise formula for these complements :
Do you agree ?
We don't need to include because it's ...
And thus the third axiom is automatically checked.
So the smallest -algebra containing A,B,C (also called the generated -algebra by A,B,C) is :