I'm trying to prove that any finite non-empty set X has the same number of subsets of even size as it has subsets of odd size but finding it quite difficult to form a rigorous argument.
I have come up with an argument that is like this:
To create any subset of X we can go through each element of X and decide whether it is in the subset or it is not in the subset, hence for each element of X we will have that it is either in the subset or is not in the subset. For any even sized subset of X there must exist an odd subset (simply include or remove an element in X) and similarly for all odd sets there must be an even set. Thus the their are an equal number of odd/even subsets of X.
I fear this isn't a good argument and would like some advice.