take a small example and then analyze
1) --->when N = 3 (Odd number)
let S = { 1,2,3 } --- > 8 subset possible of set S
X is subset of set of size 3 containing even number of elements
X = { {$\Phi$ } , {1,2} , {2,3} , {1,3} } = 4 subsets possible containing even number of elements
2) -->when N = 2 (even number)
S = {1,2} --- > 4 subset possible of set S
X = { {$\Phi$ } , {1,2} } = 2 subsets possible containing even number of elements
therefore the general formula is that if we have a sets of size = N (odd /even)
then the number of subsets with even cardinality equals the number of subsets with odd cardinality. So
this number is $\frac{1}{2}$2n=2n−1.
X = 2n−1
A = N elements
B = {0 ,1 } -- > 2 elements
Total number of function possible from set A to set B = Y = 2n
$\frac{X}{Y} = \frac{2^{n-1}}{2^{n}} = \frac{1}{2}$