Let A = 1, 2, 3, 4, 5, 6, 7, 8 . Define B = T A : either 8 T or 7 T and C = T A : the sum of all elements of T is not a multiple of 3 . Then the number of elements in the set B C is _______.

Step-by-step solution

By De Morgan's law, (B C)' = B' C' . The set B' consists of subsets T of A such that 8 T and 7 T . The set C' consists of subsets T of A such that the sum of all elements of T is a multiple of 3 . Let T = 8 S , where S 1, 2, 3, 4, 5, 6 . The sum of elements in T is 8 + sum (S) . For this sum to be a multiple of 3 , we must have 8 + sum (S) 0 3 , which implies sum (S) 1 3 . We group the elements of 1, 2, 3, 4, 5, 6 by their remainder modulo 3 : Remainder 0 : 3, 6 ( 2 elements) Remainder 1 : 1, 4 ( 2 elements) Remain