the fact that you cut S in the middle part makes all the works..
to illustrate more, if you stay in the middle and YOU are in the middle, and since you know you are in the middle, there will be equal number from your left and from you right.. that is the time when you could say that they have the same countability (cardinality).. get the point?
as per S can be irrationals, you can always write [a,b]\Q.