Discrete Mathematics What is a proof?

The pigeonhole principle is a basic counting technique. It is illustrated in its simplest form as follows: We have n + 1 pigeons and n holes. We put all the pigeons in holes (in any way we want). The principle tells us that there must be at least one hole with at least two pigeons in it. Why is that true? Try to visualize the example of n = 2; therefore, we have 3 pigeons and 2 holes. Let’s try to avoid the consequence stated by the principle. If all pigeons must be placed in holes, the first one must be placed in some hole. This hole can no longer be used. Now the second pigeon must occupy a different hole. The third pigeon must share a hole with another pigeon. It is obvious that this argument/proof can be generalized to any n. However, it is very mechanical. For instance, when presenting this proof and showing that any strategy will fail to avoid putting two pigeons in the same hole, you will start by saying something like: let’s place pigeon 1 in hole 1. One might say in response to that: but what if there is another strategy? You are going to say: well it does not matter which hole you choose for pigeon 1. So, basically you have to articulate your proof. Here’s an easier proof using a technique called proof by contradiction. In a proof by contradiction you start by the opposite of what you claim, and then try to reach something that is false (yes that’s funny!). If your logic is correct, this can only mean one thing: your starting point is false. So, what is the opposite of our claim? Our claim is that at last one hole will contain at least two pigeons. The opposite of the claim is that every hole has at most one pigeon. Assume every hole has at most one pigeon. Then the total number of pigeons is at most 1+1+. . .+1 (n times), which is n, a contradiction! Dirichlet was the first to articulate this principle in proving that for any real number α and any integer n, there exist integers p and 1 ≤ q ≤ n, such that: