Question

Let $a$ be a set of 10 distinct integers between 1 and 100 (both inclusive). Show that there are two nonempty and disjoint subsets of $a$ such that the sum of all its elements are the same.

a) Always true
b) Some× true
c) Rarely true
d) Never true

Answer 1

The statement is always true due to the pigeonhole principle, which guarantees that among the 2^10 - 1 possible nonempty subsets of a set of 10 distinct integers, at least two subsets will have the same sum and be disjoint.

Step-by-step explanation:

The student's question pertains to set theory and the properties of subsets within a given set of integers. Specifically, they are asking whether it is always true that, given a set of 10 distinct integers, there are two nonempty and disjoint subsets with the same sum. This concept revolves around the pigeonhole principle.

Based on the pigeonhole principle, there are more possible subsets (210 - 1, for nonempty subsets) than total sums achievable (maximum of 945, by summing integers from 91 to 100), thus ensuring that at least two different nonempty subsets must share the same sum. This makes the statement always true. Therefore, the correct answer to this question is option (a) Always true. The subsets are disjoint because they don't share any of the same elements while still having the same sum.