Pigeonhole Principle: An Outstanding Axiom Or Just A Common Sense?
A fundamental theory of mathematics which makes common sense sound sophisticated.
The term ‘Pigeonhole Principle’ itself indicates that this should be something connected to pigeons or pigeonholes. Okay, the answer is yes. Precisely, this principle tells us that, if (n+1) pigeons are to be nestled in n pigeonholes, at least one pigeonhole must be filled with more than one pigeon. In other words, if (mn-1) number of pigeons are to be nestled in n pigeonholes, m,n being natural numbers, i.e. m,n= 1,2…, then at least one pigeonhole must contain less than m pigeons. Now if we generalise this statement, it becomes that, if (kn+m) items are to be put in n containers with m being a natural number then at least one container has to contain more than k objects. The proof of this statement is as follows.
Let us assume that, the above statement is wrong. Then each container contains at most k objects, which implies that the number of the total objects is at most kn. This arises a contradiction to the fact of m being a natural number. Hence the pigeonhole principle.
After going through the above paragraphs, you may think what is the speciality of this? Everybody has this common sense. In fact, why is this proposition even considered as an…
