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Is there anyone who is familar with Meausre Theory?

If so, would appreciate your guidance on below problem.

Question:
Let X be a non-empty set and let P and Q be two sigma-algebras on X. Is P U Q(union of P and Q) a sigma-algebra on X?

My approach towards the question:

My intuition was that this is not true in general. So, the next idea that came to my mind is to disaprove this by using an counterexample.

As the first step, I selected a non-empty set X as follows:

X = {a, b, c}, where a, b and c are distinct.

Then I listed out all the possible sigma-algebras that came to mind, which can be derived from X, as follows:

Here, @ denote the null set.

P1 = {@, X}
P2 = { @, X, {a}, {b, c} }
P3 = { @, X, {b}, {a, c} }
P4 = { @, X, {c}, {a, b} }
P5 = { @, X, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} }

Then, I considered P2 U P3 = { @, X, {a, b}, {a, c}, {b, c} }

So, in this case, {c}, the complement of {a, b} is not included in P2 U P3, hence not agreeing with the second property of a sigma-algebra.

Is this a valid counterexample which can prove that the intial statement given in the question is not valid in general?

Yes, your approach and counterexample are correct. Your counterexample shows that the union of two sigma-algebras is not necessarily itself a sigma-algebra. In general, the union of two sigma-algebras is a sigma-algebra if and only if one of them is a subset of the other one. So, your counterexample shows that the initial statement given in the question is not always true.

Thank you!