If A ∪ B = A, what can we say about B?

• A. A is a subset of B
• B. A equals B
• C. B is a subset of A
• D. A and B are disjoint

Answer: (C)

The situation shows that nothing outside A is contributed by B. If A ∪ B equals A, then every element of B must already be in A, because elements of B are part of the union, and that union is just A. So B is contained in A, meaning B is a subset of A. It can be that B equals A, or that B is smaller than A, but there’s no element of B outside A. Disjointness isn’t implied unless B is empty, which is a special case, and equality isn’t required.