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(Q). Consider a set x = {1, 2, 3, …….., 9, 10}. What is the number of pair {A, B} such that A ⊆ x and B ⊆ x also A ≠ B and A ∩ B = {2, 3, 5, 7}.
Solution:-

Since it is given that A ∩ B = {2, 3, 5, 7}, so remaining numbers {1, 4, 6, 8, 9, 10} have three option to go in either set A or set B or neither in set A nor in set B. So each member of set {1, 4, 6, 8, 9, 10} has 3 option of selection. So total number of ways is 3⁶ but since A ≠ B, so subtract the case when A = Φ & B = Φ i.e. A = B.
Hence required number of ways = 3⁶ – 1     Answer