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) | |