Let X = 1, 2, 3, 4, 5, 6, 7, 8 , Y = 1, 2, 3, 4 and Z = 3, 4, 5, 6 . The number of subsets C of X such that C Y and C Z is ______

Step-by-step solution

The total number of subsets of X is 2^ n(X) = 2^8 = 256 . We need to find the number of subsets C that have a non-empty intersection with both Y and Z . We can use the Principle of Inclusion-Exclusion to find the number of subsets that fail this condition. A subset C fails the condition if C Y = or C Z = . Case 1: C Y = This means C must be a subset of X Y = 5, 6, 7, 8 . The number of such subsets is 2^4 = 16 . Case 2: C Z = This means C must be a subset of X Z = 1, 2, 7, 8 . The number of such subsets is 2^4 = 16