Question:medium
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 :Cߏ∩B ≠ φ} is _____________.
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 :Cߏ∩B ≠ φ} is _____________.
Updated On: Apr 13, 2026
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Correct Answer: 112
Solution and Explanation
Let the set C be a subset of A where C ∩ B ≠ ∅. This means C should have at least one element from B. To solve, first calculate total subsets of A. If A has n elements, the number of subsets is 2ⁿ. Here, n = 7 (elements in A), so total subsets = 2⁷ = 128.
Now, find the number of subsets of A where C ∩ B = ∅. For this, consider the elements not in B: {1, 2, 4, 5}. The number of subsets of these elements (since each element can independently be in or out of C) is 2⁴ = 16.
To find subsets where C ∩ B ≠ ∅, subtract subsets where C ∩ B = ∅ from total subsets: 128 - 16 = 112.
Therefore, the number of subsets C such that C ∩ B ≠ ∅ is 112, which fits the range 112,112.
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