-A- way of comparing quantities is a 1-1 correspondence.

If you have another and it is logically consistent, then use it. As long as 2 + 2 = 4 it is cool.

y = x if x even, x+1 if x odd. That is a 2-to-1 map/function/correspondence, which implies the set X (all integers) has twice as many numbers as the set Y (even integers).

That is ill defined. What are two sets and what is the mapping.;

Let the two sets be A and B, both subsets of the integers.

Now let x in A. If x is even you say f(x) = x

and if x is odd you say f(x) = x + 1.

if x = x' then f(x) = f(x') now let us check the -inverse-

suppose f(x) = f(x') does this imply x = x'. No

Example f(4) = 4 but f(3) = 4 so the inverse function is not well defined.

So, 4 and 3 which are not equal get mapped into the same element. bzzzzzzzt. Wrong.

So your function as you defined it is NOT a 1-1 onto mapping. It has an ill defined inverse.

Go back and try something else.

or take a course in algebra or set theory.

Do something useful.

Ba'al Chatzaf