By "no more than elements", perhaps you mean "at least elements"? Because for instance if your set contains less than elements, you certainly can't pick a subset of it having elements.
Please check the exact statement of the problem!
There is a natural number . Prove that from any set consisting of integer numbers that has more than elements we can take a subset S that has elements and:
For any two different subsets the sum of all elements of set is different from the sum of all elements of set . (We assume that the sum of all elements of an empty set equals 0).
I would be really grateful if anyone could help me with this assignment.
I've already tried set theory and induction. Now I think this problem must have something to do with number theory, but this branch is so vast that I don't know where to start looking.