How do I know what quantity systems are part of sigma algebra?

Question

Given is the set $Ω = \left \{3,4,5,6\right \}$ .

How do I know what quantity systems correspons to an event algebra?

The definition is any collection that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections.

The first example meets these conditions, the second does not ({3,5} is not included). I can eliminate the third family immediately, it doesn’t contain {3, 4, 5, 6}. And the last example meets these conditions. So is this all good?

Only first and last example meet these conditions.

Answer 1

Yes, you are right. Only the first and the last set are $\sigma$-algebras. Note that you do not need to check the intersections, as this follows from unions and complements.