Question: Let S = {2, 3,5,7, 11, 13, 17, 19} be the set of prime numbers less than 20. If A is a subset of S, we can form the sum and product of the elements of A. For example, if A = {7, 11, 13}, then the associated sum is 7 + 11 + 13 = 31 and the associated product is 7(11)(13) = 1001.
(a) Use the Pigeon-Hole Principle to show that there are four nonempty subsets of S with the same sum.
(b) Are there two nonempty subsets of S with the same product? Explain.