Let S = 1, 2, 3, 4, 5 and P(S) be the power set of S . Let f(A) denote the sum of all elements in a subset A of S , with f( ) = 0 . A relation R is defined on P(S) such that A , R , B if and only if f(A B) = f(A) + f(B) . Then the relation R is :

Step-by-step solution

For any two subsets A and B , the sum of elements satisfies the inclusion-exclusion principle: f(A B) = f(A) + f(B) - f(A B) The given relation A , R , B requires f(A B) = f(A) + f(B) . Substituting this into the principle yields f(A B) = 0 . Since all elements of S are strictly positive integers, the sum of elements in A B can be zero if and only if A B = . Thus, A , R , B is equivalent to saying A and B are disjoint sets. For reflexivity: A A = A . For A , R , A to hold, we need A = . Since this is not true for a