Answers

2016-07-17T19:31:51+08:00
In mathematics, the power set (or powerset) of any set S, written P(S), ℘(S), P(S), ℙ(S) or 2S, is the set of all subsets of S, including the empty set and S itself. :0
0
2016-07-17T19:41:48+08:00
In mathematics, the power set (or powerset) of any set S,n P(S), ℘(S), P(S), ℙ(S) or 2S, is the set of all subsets of S, including the empty set and S itself. Example: The power set is the set of all subsets of a given set. For the set S = {1,2,3} this means subsets with 0 elements: 0 (the empty set) subsets with 1 element: {1}, {2}, {3} subsets with 2 elements: {1,2}, {1,3}, {2,3} subsets with 3 elements: S Hence: P(S) = {0, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, S} Therefore, we have: card(S) = 3 and card(P(S)) = 8 = 23 For the set S = {1,2,3,4} this means: subsets with 0 elements: 0 (the empty set) subsets with 1 element: {1}, {2}, {3}, {4} subsets with 2 elements: {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4} subsets with 3 elements: {1,2,3}, {1,2,4}, {1,3,4} {2,3,4} subsets with 4 elements: S Therefore, we have: card(S) = 4 and card(P(S)) = 16 = 24 Finally, if S = {1,2,3,4,5,6} then, based on the above examples, we would suspect that card(S) = 6, therefore card(P(S)) = 26 = 64 In fact, if a set S contains n elements, then its power set will contain 2n elements. This can be proved by induction as an exercise.
0