I want the detail answer

1. Let A be the set of alternatives and N be the set of citizens. Strict preference of citizen 2' 6 N over A is denoted by Pi. (a) Majority rule: a E A is socially as good as b E A 4:} |{i E N | aPib}| 2 W e N | bean (b) Approval voting rule: Fix an integer, 1 S k S |A|. Take a E A andz' E N , 3(a, i) = l ifz' ranks a among top k alternatives, otherwise s(a,z') = 0. a E A socially as good as b E A 4:) ZEN s(a,i) Z ZEN 3(1), 2) (c) Reverse dictatorship: Fix k E N. a E A is socially preferred to b E A (l) bPka Check whether the above social rankings satisfy transitivity, HA and Praeto