Let the subset be denoted as a n-bit sequence of 0s and 1s
A 0 at the ith position indicates absence of the ith element
A 1 at the jth position indicates presence of the jth element
For example, let the set be {1,2,3}
Then {1,2} can be represented as 110
{1,2,3} can be represented as 111
and so on
Now the number of subsets of even cardinality in an n-element set = number of n bit strings with an even number of ones
number of n bit strings with an even number of ones = number of n bit strings with an odd number of ones
=> number of n bit strings with an even number of ones = (number of different n bit strings)/2 = $2^{n-1}$
