Question

Prove that for any collection of \(n\) people, two persons have the exact same number of acquaintances in the group provided that each person has at least one acquaintance.

Short answer

Two people have the same number of acquaintances by the Pigeonhole Principle, because only 1 to n-1 acquaintances are possible for n people.

Step-by-step solution

Understand the Problem

We need to prove that in a group of \(n\) people, where each has at least one acquaintance, there are at least two people with the same number of acquaintances.

Apply the Pigeonhole Principle

Label each person as \(P_1, P_2, \ldots, P_n\). Any person can have from 1 to \(n-1\) acquaintances. If everyone had different numbers of acquaintances, the people would be labeled 1 to \(n-1\). Thus, there would be \(n-1\) possible labels for \(n\) people, which contradicts our initial condition that everyone has at least one acquaintance (since someone should have acquaintance '0' which isn't supported).

Conclude Using a Contradiction

Since it is impossible for everyone to have different numbers of acquaintances as they range from 1 to \(n-1\) (\(n\) people in \(n-1\) slots), by the Pigeonhole Principle, at least two people must have the same number of acquaintances.

Key concepts

Understanding Graph Theory

Graph theory is a branch of mathematics that studies the properties and relationships of graphs. A graph is a collection of points, known as vertices, connected by lines, which are referred to as edges. In our exercise, each person is treated as a vertex, and an edge represents an acquaintance relationship between a pair of people. Understanding this, we translate the problem of finding shared acquaintances into a graph problem.

• Vertices: Each vertex denotes a person in the group.
• Edges: If two people know each other, an edge connects their respective vertices.
• Degree: The number of edges connected to a vertex. It represents the number of acquaintances a person has.

In our case, the degree of each vertex varies from 1 to n-1, making the study of vertex degrees crucial in solving the problem. Graph theory provides a visual and structured way to comprehend complex relationship networks, like those involving people and their acquaintances.

Fundamentals of Combinatorics

Combinatorics deals with counting, arrangement, and combination problems. Within this field, the Pigeonhole Principle is a simple yet powerful technique often used. It states that if you have more "pigeons" than "pigeonholes," at least one pigeonhole must contain more than one pigeon.

In our problem, apply this principle where:

• "Pigeons" are the people.
• "Pigeonholes" are the possible numbers of acquaintances each person can have (ranging from 1 to n-1).

Since the number of possible different acquaintances (1 to n-1) is one less than the number of people ( ), at least two people must inevitably end up with the same number of acquaintances. This insight seamlessly guides us toward the conclusion we need.

Exploring Proof Techniques

Proof techniques are essential for establishing the truth of mathematical statements. In this exercise, we employ the contradiction method, which is a common proof technique. It involves assuming that a statement is false, and then demonstrating that this assumption leads to a contradiction.

Here's how it works in our problem:

• Assume that everyone has a different number of acquaintances.
• This implies we could label people with numbers 1 to n-1.
• However, because there are n people but only n-1 possible labels, one label must repeat.
• Thus, the assumption leads to a contradiction.

By exposing such contradictions, proof by contradiction offers a compelling way to affirm the validity of a proposition. It confirms that at least two individuals in the group must have the same number of acquaintances, aligning with the implications of the Pigeonhole Principle.