Question

Discrete math : Show your work please.

Consider a set X of 10 positive integers, none of which is greater than 100. Show that it has two distinct subsets whose elements have the same sum.

Answer 1

let a set x of 10 positive integers will be two digits. now claim that it has two distinct subsets whose elements have the same sum. Note that there are distinct subsets of our set of 10 two-digit numbers. Also note that the sum of the elements of any subset of our set of 10 two-digit numbers must be between 10 and , which is less than . There are even less attainable sums. The Pigeonhole Principle then implies that there are two distinct subsets whose members have the same sum. Let these sets be and . Note that and are two distinct sets whose members have the same sum. These two sets are subsets of our set of 10 distinct two-digit numbers, so this proves the claim.