{A,B,C} is of greater cardinality than {A,B}. However, the argument that "a rule works for finite numbers, so it must work for infinite numbers" is clearly false.
The distance between 0 and 1 is smaller than the distance between 0 and 2. The number of points between 0 and 1 is larger than the number of rational numbers.
For a your mapping, see: http://math.stackexchange.com/questions/512397/is-there-a-si...