No, f need not to be one-one. Right now I am in a little rush I will give the counter-example later on.
I read your comment in gate 1988 there you wrote that S is equipotent to itself and it is given that S is surjective so, it has to be injective.
Look A and B are said to be equipotent sets if there exists a bijective mapping from A to B rt.
Yes,you are right that S is equipotent to itself because there exists a bijective mapping from S to S but that does not mean all mappings that can be defined from S to S has to be bijective. There may exist some mapping between S to S which is not bijective.
I hope you are getting my point.