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Let A = 1, 2,...

Question

Let $A = {1, 2, 3, 4, 5, 6, 7}$ and $B = {3, 6, 7, 9}$ . Then the number of elements in the set ${C \subseteq A : C \cap B \neq ϕ}$ is

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Solution

As $C \cap B \neq ϕ, C$ must be not be formed by ${1, 2, 4, 5}$ and $C$ should not conatain $9$
$∴$ atleast one element from ${3, 6, 7}$ and from ${1, 2, 4, 5}$ without any restriction
So required number of eleents will be $(2^{3} - 1) \cdot 2^{4} = 112$

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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

As C∩B≠ϕ, C must be not be formed by {1,2,4,5} and C should not conatain 9 ∴ atleast one element from {3,6,7} and from {1,2,4,5} without any restriction So required number of eleents will be (23−1)⋅24=112

Let $A = {1, 2, 3, 4, 5, 6, 7}$ and $B = {3, 6, 7, 9}$ . Then the number of elements in the set ${C \subseteq A : C \cap B \neq ϕ}$ is

As $C \cap B \neq ϕ, C$ must be not be formed by ${1, 2, 4, 5}$ and $C$ should not conatain $9$
$∴$ atleast one element from ${3, 6, 7}$ and from ${1, 2, 4, 5}$ without any restriction
So required number of eleents will be $(2^{3} - 1) \cdot 2^{4} = 112$