# Prove Nonempty Subset Sum of 5 Distinct Single-Digit Integers

• liahow
In summary, the conversation discusses the meaning and importance of proving the existence of a nonempty subset of five distinct single-digit integers whose sum equals a given number. The "nonempty" and "distinct" conditions ensure the usefulness and uniqueness of the subset. Proving the existence of such a subset can have practical and theoretical significance and can be extended to larger sets of integers with adjusted conditions. Various approaches can be used to prove the existence of such a subset, such as proof by contradiction or constructive proof.
liahow
No idea where to start with this one...to prove that it is not possible to have a set B of 5 distinct positive single-digit integers such that every possible nonempty subset of B has a different sum. How do I approach it/do it?

Given 5 distinct numbers, there are 2^5 - 1 possible sums of subsets, excluding the sum of all 5 which is 31.

However, the biggest possible sum of 4 distinct single digit numbres is 9+8+7+6= 28, so apply the pigeon hole principle.

I'll try that, thanks for the suggestion!

## 1. What does "Prove Nonempty Subset Sum of 5 Distinct Single-Digit Integers" mean?

The phrase "Prove Nonempty Subset Sum of 5 Distinct Single-Digit Integers" refers to a mathematical problem that asks for a proof that there exists a nonempty subset (a subset with at least one element) of five distinct single-digit integers whose sum is equal to a given number.

## 2. Why is it important to prove the existence of such a subset?

Proving the existence of such a subset has both practical and theoretical significance. On a practical level, it can help in solving real-world problems that involve finding combinations of numbers that add up to a specific target. On a theoretical level, it can contribute to our understanding of number theory and combinatorics.

## 3. What is the relevance of the "nonempty" and "distinct" conditions?

The "nonempty" condition ensures that the subset we are looking for has at least one element, as an empty subset would not be useful in solving the problem. The "distinct" condition ensures that each integer in the subset is unique, preventing any repeats that could lead to multiple solutions or make the problem too easy.

## 4. How would you go about proving the existence of such a subset?

There are several approaches that can be used to prove the existence of a nonempty subset sum of 5 distinct single-digit integers. One method could be to use a proof by contradiction, assuming that no such subset exists and then showing that this leads to a contradiction. Another approach could be to use a constructive proof, where a specific example of such a subset is provided.

## 5. Can this problem be extended to larger sets of integers?

Yes, this problem can be extended to larger sets of integers, with the conditions adjusted accordingly. For example, "Prove Nonempty Subset Sum of 10 Distinct Single-Digit Integers" would ask for a proof that there exists a nonempty subset of 10 distinct single-digit integers whose sum is equal to a given number. However, as the set size increases, the complexity of the problem also increases, making it more difficult to prove the existence of a solution.

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