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Chapter 1: Problem 43

Freedonia has 50 senators. Each senator is either honest or corrupt. Suppose you know that at least one of the Freedonian senators is honest and that, given any two Freedonian senators, at least one is corrupt. Based on these facts, can you determine how many Freedonian senators are honest and how many are corrupt? If so, what is the answer?

Short Answer

Expert verified
There is 1 honest senator and 49 corrupt senators.

Step by step solution

01

- Understand the Problem

There are 50 senators. Each is either honest or corrupt. At least one senator is honest, and given any two senators, at least one is corrupt.
02

- Analyze the Implications

If at least one senator is honest and given any two senators, at least one is corrupt, this implies that there cannot be two honest senators. Otherwise, there would be a pair of senators where neither is corrupt.
03

- Identify the Number of Honest Senators

Since there cannot be more than one honest senator but there must be at least one, there is exactly one honest senator.
04

- Determine the Number of Corrupt Senators

With 50 senators total and only one being honest, the remaining 49 senators must be corrupt.

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Key Concepts

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

Logic and Reasoning
To solve problems in discrete mathematics, it's essential to use logic and reasoning effectively. In this exercise, we start by understanding the given conditions:
  • There are 50 senators.
  • Each senator is either honest or corrupt.
  • At least one senator is honest.
  • Any two senators always include at least one corrupt senator.
Using these facts, we apply logical reasoning. Since any two senators always include at least one corrupt senator, there cannot be two honest senators. If there were two honest senators, the condition would be violated. This logical step is crucial in narrowing down the possibilities, leading to the realization that there must be exactly one honest senator.
Mathematical Analysis
Mathematical analysis involves breaking down a problem into smaller, more manageable parts and examining the implications of each part. Let's analyze the given conditions in detail:
  • Step 1: Identify that there are 50 senators.
  • Step 2: Understand that each senator is either honest or corrupt.
  • Step 3: Recognize that at least one senator is honest.
  • Step 4: Realize that any pair of senators includes at least one corrupt senator.
Combining these observations, we deduce that there can be at most one honest senator, as having more than one would contradict the condition of always having at least one corrupt senator. Finally, since we must have at least one honest senator and there are 50 senators altogether, we conclude that there is exactly one honest senator and the remaining 49 are corrupt.
Problem Constraints
Understanding problem constraints is key to finding accurate solutions. The constraints here set specific boundaries for the number and types of senators:
  • There are 50 senators in total, limiting the scope of the problem.
  • Each senator falls into one of two categories: honest or corrupt.
  • At least one senator must be honest.
  • Any two senators must include at least one corrupt senator.
These constraints guide us to the solution. Given the constraint that any pair includes at least one corrupt senator, it's impossible for any two senators to both be honest, which immediately limits the number of honest senators to one. This constraint combined with the total number of senators tells us that we must have 49 corrupt senators remaining. Understanding and applying these constraints effectively leads us to the correct solution in a structured manner.

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