Power Set

 If A is a set having n no. of elements, then power set P(A) of A is defined as the set of all subsets of A. No. of elements in P(A) is 2n. P(A) = i=nΣi=0 C(n,i) = 2n Cardinality (no. of elements) of a power set is always greater than the no. of elements in the original set. 2n > n Power Set includes an empty  set and the original set. Power Set of any set becomes an Abelian group under the operation/composition of symmetric difference with the empty set as the neutral element of the Group. If n is an odd no. and n = 2k+1 where k=0,1,2,3........ then P(A) = i=nΣi=0 C(n,i) =  i=kΣi=0 2*C(2k+1,i) If n is an even no. and n = 2k where k=0,1,2,3,....... then   P(A) = i=nΣi=0 C(n,i) =  i=kΣi=0 2*C(2k,i) - C(2k,k) Generalising the above for any n where n = 2k + r where k=0,1,2,3,....... and r is the remainder and is either 0 or 1, P(A) = i=nΣi=0 C(n,i) =  i=kΣi=0 2*C(2k+r,i) - (1-r)C(2k,k)

 No. of Elements in Set A--------------(n) No. of Elements in Powerset P(A) ------( n' = 2n ) k = r = 2k+r (1-r)C(2k,k) NO. OF SUBSETS OF THE POWER SET HAVING NUMBER OF ELEMENTS AS UNDER no. of elements in subset zero one two three four five six seven eight nine ten total Formula C(n,0) C(n,1) C(n,2) C(n,3) C(n,4) C(n,5) C(n,6) C(n,7) C(n,8) C(n,9) C(n,10) i=nΣi=0 C(n,i) no. of subsets Example-{1,2,3,4,5} {} {1},{2},{3},{4},{5} {1,2},{2,3},{3,4},{4,5},{1,5},{1,3},{3,5},{2,5},{1,4},{2,4} {1,2,3},{2,3,4},{3,4,5},{1,4,5},{1,3,4},{2,4,5},{1,3,5},{1,2,4},{2,3,5},{1,2,5} {1,2,3,4},{2,3,4,5},{1,3,4,5},{1,2,3,5},{1,2,4,5} {1,2,3,4,5} ---- ---- --- --- --- 32