Assume, by contradiction, that f is not 1-1 between subsets of with even cardinality and those of odd cardinality. Then, there exist sets with even cardinality, such that and .

Now, separate this into three cases:

1. , contradiction.

2. (same when n is in B and not A).

3.

I did the first one for you, now see if you can do the others (they are just as simple).