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 2^{n}. P(A) = ^{i=n}Σ_{i=0} C(n,i) = 2^{n} | |
Cardinality (no. of elements) of a power set is always greater than the no. of elements in the original set. 2^{n }> 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=n}Σ_{i=0} C(n,i) = ^{i=k}Σ_{i=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=n}Σ_{i=0} C(n,i) = ^{i=k}Σ_{i=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=n}Σ_{i=0} C(n,i) = ^{i=k}Σ_{i=0} 2*C(2k+r,i) - (1-r)C(2k,k) | |