Question

If A and B are the two sets containing 3 and 6 elements respectively, then what can be the maximum number of elements in A∪B?

• A. 9
• B. 10
• C. 11
• D. 12

Answer 1

The Correct Option is A

Let A and B be two sets with |A| = 3 and |B| = 6. 

The number of elements in the union of A and B is given by the principle of inclusion-exclusion: |A U B| = |A| + |B| - |A ∩ B|. 
The maximum number of elements in A U B occurs when the intersection of A and B is minimal.

Case 1: A and B are disjoint sets, i.e., A ∩ B = ∅. In this case, |A ∩ B| = 0. |A U B| = |A| + |B| = 3 + 6 = 9

Case 2: A is a subset of B, i.e., A ⊆ B. In this case, A ∩ B = A, so |A ∩ B| = |A| = 3. |A U B| = |A| + |B| - |A ∩ B| = 3 + 6 - 3 = 6

Case 3: A and B have some elements in common. Let |A ∩ B| = k. 
Since A ∩ B ⊆ A, we have 0 ≤ k ≤ |A| = 3. Since A ∩ B ⊆ B, we have 0 ≤ k ≤ |B| = 6. Thus, 0 ≤ k ≤ 3. |A U B| = |A| + |B| - |A ∩ B| = 3 + 6 - k = 9 - k. 
To maximize |A U B|, we need to minimize k. 
The minimum value of k is 0. |A U B| = 9 - 0 = 9.

Therefore, the maximum number of elements in A U B is 9.