Application of PMI in real life and branches of Mathematics

The principle of Mathematical Induction (PMI) is deductive reasoning which often uses logic, assumptions, and contradictions to reach conclusive proof. The focus of Mathematical Induction has a lot of significance in real life. We can use it to test a given statement by assuming a situation to be accurate and reaching a conclusion by drawing logical inferences from similar problems. For example, mathematical induction is generally used to prove that statements are factual of all natural numbers. The inductive approach is first to verify that the information posted is accurate for any positive natural number and then prove that the same must also be true of the following number. It has wide applications and can also be used in real-life scenarios.

There are several examples of mathematical induction in real life:

1) The most famous and vivid example is toppling dominoes.

You must have seen various instances wherein a single object can initiate a chain reaction in real life. Dominoes are a great example of it. If you place n number of dominoes in line with each other at a distance where one domino will topple its next domino, it will level all the dominoes. But in mathematical induction, you have first to test whether the first two dominoes will topple each other or not. Once you prove that the distance at which both dominoes are enough to topple the other if the first domino falls, it is assumed that all the other dominoes are at the same distance from each other, it would be proved that all dominoes will fall.  

2) The sinking of the Titanic

It is a real-life example of mathematical induction. When the titanic ship hit the massive iceberg, it started flooding its lower quarters at the front bulkhead. If we go by PMI, the boat would be wholly attacked. If it is tested that the given bulkhead would in all possibility keep flooding, then it would be true that once the front bulkhead gets completely flooded, the next bulkhead would undergo the same fate, thus attacking the whole ship.

3) Zipping a zipper successfully

This is similar to the falling domino problem. In this situation, too, if you can successfully zip or unzip the first tooth zipper, the induction base would be genuine. Now, the next teeth of the zipper must also follow the same rule for a successful knot. Henceforth, if the kth zip is successfully closed, the zipper will complete the next teeth successfully. The value of k can be any value between the first and last teeth of the zipper. Thus, it can be proved that the zipper will be successfully zipped.

4) The common knowledge (logic)

Problem:

Assume that an isolated island has a fixed population. Out of these, k number of people have red eyes, and the rest of the people have green eyes. No one on the island knows their eye color but knows every other person on the island. If a person on the island ever discovers they have blue eyes, that person would be outcast and leave on the first night. If it cannot be determined, they can stay on the island. There are no reflective surfaces, and there is no communication of eye color.

One day, some foreigners anchor their ship on the island. They call all the islanders and announce that at least one of them has red eyes. Naturally, the announcement is assumed to be true since it is common knowledge that foreigners are truthful. Following this logic, it is thus common knowledge that at least one islander has red eyes.

Solution

The straightforward answer to the above problem would be that all the red-eyed people will be outcast on the kth night after the announcement.

As for how? By following the principle of mathematical induction, if k = 1, there is precisely one red-eyed islander; the person will realize that he alone has red eyes since he already knows the eye color of every other islander. If k = 2, then with the principle of common knowledge, no one will leave on the first night. The two red-eyed people, seeing only one person with red eyes and that no one left on the 1st night (and thus that k > 1), will go on the 2nd night. Through the principle of induction, it can be reasoned that no one will leave at the first k − 1 night if and only if there are at least k red-eyed people. Those with red eyes, seeing k − 1 red-eyed people among the others and knowing there must be at least k, will conclude that they must have red eyes and leave.

In this particular scenario, the common knowledge principle comes into effect. The common knowledge principle is nothing more or less than what the islanders already know. Meaning, when the foreigners announce that at least one of them has red eyes, it becomes common knowledge since there was no way for the islanders to communicate the same. Hence, before this fact is announced, the truth is not comprehensible by the islanders; hence is not common knowledge. But once it is conveyed to them by the foreigners, it becomes common knowledge. Thus a fact that becomes of common understanding has an evident and noticeable effect. When the foreigners publicly announce a presence already known to all but not evident becomes common knowledge, the red-eyed people on the island eventually deduce their status and leave.

As we have seen, mathematical induction explains real-life scenarios logically. It is an essential part of algebra and other parts of mathematics and helps solve critical questions that we experience in daily life through test, assumption, proof, and explanation.

Read about the recent developments in mathematical induction. Find out more on related topics such as Applications of PMI in Proving Divisibility Rules and relation of Fibonacci sequence with installation, and many more.