Let A={1,2,3,4,5,6,7} and B={3,6,7,9}. Then the number of elements in the set {C ⊆ A: C ∩ B ≠ φ} is ___________.

Solution

Maths sir · 4 steps

Count the subsets that miss B completely, then subtract them from all subsets of A.

1. Find the overlap. Only the elements of A that are also in B matter here. A ∩ B = {3, 6, 7}, because 9 is not in A.
2. Count all subsets. A has 7 elements, so it has 2⁷ = 128 subsets in total.
3. Remove the bad ones. A subset C has C ∩ B = φ exactly when it avoids 3, 6 and 7. Such a C is built only from {1, 2, 4, 5}, so there are 2⁴ = 16 of them.

   2⁷ - 2⁴ = 128 - 16 = 112

4. So. The number of subsets of A that meet B is 112.

The common slip: 120. Removing all 4 elements of B from A leaves 3 elements, giving 128 - 8 = 120. But 9 is not in A, so only 3 elements go and 4 remain.

The idea to keep. For "at least one" conditions on subsets, count the complement: all subsets minus those that avoid the given elements. Only elements actually inside A count.

Answer from the NTA final answer key. Solution by PrepWiser.