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Chapter 6: Problem 25

(a) Let \(A\) be a set of seven (distinct) natural numbers none of which exceeds 21 . Prove that the sums of the elements in all the nonempty subsets of \(A\) are not distinct. (b) Improve the result of (a) by showing that the result holds under the assumption that the integers of \(A\) do not exceed 23 . (c) Assume none of the elements of \(A\) exceeds \(12 .\) At least how many subsets of \(A\) must have the same sum?

Short Answer

Expert verified
(a) Some subsets have the same sum. (b) Same result for numbers not exceeding 23. (c) At least 2 subsets share a sum.

Step by step solution

01

Understand the problem

We are given a set \(A\) with seven distinct natural numbers, and we need to show that among all possible non-empty subsets, there are subsets with identical sums when the largest number allowed in \(A\) is 21, then 23. Further, we look for the number of subsets with identical sums when the largest number is 12.
02

Calculate possible sums of subsets

The number of non-empty subsets of a set with seven elements is \(2^7 - 1 = 127\). For part (a), since the maximum sum of elements in any subset is less than or equal to \(1 + 2 + ... + 21 = 21\times 7 = 147\), there are far fewer potential distinct sums than there are subsets, ensuring some subset sums must coincide.
03

Apply pigeonhole principle for (a)

For part (a), the pigeonhole principle applies because the number of possible subset sums (ranging from 1 to 147) is less than the number of subsets, 127, ensuring that at least two subsets have identical sums.
04

Extend to larger maxima in (b)

When none of the numbers in \(A\) exceed 23, the maximum possible sum of the elements of any subset still fits within a restricted range since \(23\times 7 = 161\). Thus, having 127 subsets correlate with fewer than 161 possible distinct sums still necessitates duplicated sums (again using the pigeonhole principle).
05

Redefine the constraints in (c)

If numbers in \(A\) do not exceed 12, then the maximum sum of all elements is \(1 + 2 + ... + 12 = 78\). As there are 127 possible subsets but only 78 possible distinct sums, by the pigeonhole principle, at least 2 (or more if we distribute evenly) subsets must share the same sum.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Subset Sum Problem
The Subset Sum Problem is a fascinating challenge in mathematics and computer science. It asks whether there exists a subset of numbers that adds up to a particular target number. In the context of this exercise, you are asked to find subsets within a set that have identical sums.
  • When comparing the number of subsets (127) to possible sums, it becomes evident why some sums would repeat. There are simply too many subsets compared to the potential distinct sums.
  • This exercise helps to showcase how even small sets of numbers can quickly grow complex, highlighting the importance of mathematical intuition and instinct.
Though it typically involves integers, this problem has real applications in cryptography and optimization problems, where solutions need to be quick and resource-efficient.
Discrete Mathematics
Discrete Mathematics deals with distinct and separate elements, often focusing on countable structures. This includes study and application in sets, graphs, and logical statements.
  • The subset sum problem, tackled in this exercise, is a key topic in Discrete Math, due to its reliance on the properties of discrete structures like sets and integers.
  • Techniques often used include recursive exploration, iterative mechanisms, and combinatorial analysis to find solutions.
Part (a) and (b) use this foundational knowledge, applying principles like the Pigeonhole principle, which is itself a critical idea in discrete problem-solving. Discrete Mathematics underpins much of modern theoretical computer science and applied logic.
Natural Numbers Set
The set of natural numbers consists of counting numbers: 1, 2, 3, and so on. These numbers are simple yet powerful in constructing mathematical theories and practical solutions.
  • In this exercise, natural numbers are pivotal as elements of set A, capped at particular values. These caps are deliberate, providing room for mathematical exploration.
  • By limiting these numbers, as in parts (a), (b), and (c), diverse mathematical principles鈥攅specially additive aspects鈥攃an be observed and analyzed deeply.
Natural numbers form a foundational building block of mathematics, essential in both basic arithmetic and more advanced disciplines like number theory.
Combinatorics in Number Theory
Combinatorics involves counting, arranging, and optimizing configurations of objects. When applied to number theory as in this problem, it offers rich insight into number structures and properties.
  • In part (a) and (b), combinatorial principles help determine which subsets have identical sums. These situations reveal patterns and repetitive structures.
  • Combinatorics also supports understanding how subsets can be distributed to ensure duplication, enabling solutions to complex problems like the subset sum problem.
Utilizing these principles, mathematicians and scientists can better approach problems involving finite and countable cases, deducing clear outcomes from apparent disorder.

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Most popular questions from this chapter

Given any positive integer \(n\), show that some multiple of \(n\) is an integer whose representation in base 10 requires just \(3^{\prime} \mathrm{s}\) and \(0^{\circ} \mathrm{s}\). \(\left[\right.\) Hint : Let \(M_{1}=3, M_{2}=33, M_{3}=333, \ldots\). Then think about the remainders when each of these numbers is divided by \(n .]\)

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