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Let In total there are subsets of .
If is odd then there is a one-to-one correspondence between sets with even cardinality and sets with odd cardinality. Subset corresponds with its complement . Consequently, the number of subsets with even cardinality equals the number of subsets with odd cardinality. So this number is
If is even then put one element aside. With the trick described above we find that has subsets with even cardinality and also subsets with odd cardinality. The subsets of with odd cardinality become subsets with even cardinality if element is added to each of them. This gives us subsets of with even cardinality. So in both cases the number of subsets of of even cardinality equals the number of subsets of of odd cardinality.