Problem 5 - Discrete Mathematics with Applications

Question

Show that in any group of eight people, at least two must have been born on the same day of the week.

Step-by-Step Solution

Answer

By applying the pigeonhole principle, we have 8 people (pigeons) and 7 days of the week (pigeonholes). Since the number of pigeons (8) is greater than the number of pigeonholes (7), we can conclude that there must be at least one day of the week with more than one person born on it. Therefore, in any group of 8 people, there will always be at least two people who share the same birthday (were born on the same day of the week).

Step 1: Identify the pigeonholes and pigeons

In this problem, the pigeonholes represent the days of the week (Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday) for a total of 7 pigeonholes. The pigeons represent the people in the group, and there are 8 people.

Step 2: Apply the pigeonhole principle

The pigeonhole principle states that if n items (pigeons) are put into m pigeonholes with n > m, then at least one hole must contain more than one item. In our case, we have 8 pigeons (people) and 7 pigeonholes (days of the week). Since the number of pigeons (8) is greater than the number of pigeonholes (7), we can apply the pigeonhole principle, which tells us that there must be at least one pigeonhole (day of the week) with more than one pigeon (person).

Step 3: Draw the conclusion

By applying the pigeonhole principle, we can conclude that in any group of 8 people, there will always be at least two people who share the same birthday (were born on the same day of the week).

Key Concepts

Discrete Mathematics

Discrete Mathematics is a branch of mathematics that deals with objects that can assume only distinct, separated values. It encompasses various topics including logic, set theory, graph theory, and combinatorics, among others. When studying discrete mathematics, reasoning and problem solving are essential, especially when dealing with finite or countable sets.

Within discrete mathematics, the pigeonhole principle is an important concept widely used to solve problems related to counting and allocation. It is a form of mathematical logic that applies when there are more ‘objects’ than ‘containers’. These objects and containers are figuratively referred to as pigeons and pigeonholes, respectively. In the given exercise, we utilize this principle to prove a relatively simple fact about daily life – considering the days of the week as pigeonholes and people as pigeons.

Combinatorics

Combinatorics is the study of countable discrete structures and which falls under the umbrella of discrete mathematics. It involves counting the number of ways certain configurations can occur, which makes it very useful in the context of probability and constructing arrangements. One of the key topics within combinatorics is the pigeonhole principle, which provides a straightforward method of proof for certain types of problems.

The power of combinatorics is evident when approaching problems that at first glance seem overwhelming due to the vast number of possibilities. By using combinatorial reasoning, we can simplify these problems by systematically counting or grouping possibilities in a logical way. The challenge provided in the exercise can be classified as a combinatorial problem since it asks for an arrangement (birthdays) among a set of objects (people) based on a constraint (days of the week).

Proof Techniques

Proof techniques are fundamental tools across all fields of mathematics for establishing the truth of mathematical statements. Discrete mathematics, in particular, relies heavily on a variety of proof methods such as direct proof, proof by contradiction, induction, and the pigeonhole principle, among others. A proof essentially serves as a logical argument that demonstrates why a given proposition is always true.

In solving the exercise, we use a direct application of the pigeonhole principle as our proof technique. By straightforward logical analysis – comparing the number of people (8) to the days of the week (7) – we directly prove that the claim 'at least two people in any group of eight will share a birthday' is an inevitable truth. This elegant solution bypasses any need for complex calculations or assumptions, showcasing the efficiency of using such proof techniques in discrete mathematics.