In a stadium, 50000 strong spectators are watching a cricket game. There must be at least two people who were born on the same day of the same year. Are we certain? Can we prove it? Yes, there is a way.
Let us assume that everyone in the crowd is aged between 0 and 80. In each year there are 365 days, so, in 80 years there are no more than 29200 different birth dates. But the crowd is bigger than 29200, then there is certain to be a duplicate. We can be 100% sure.
This is called "pigeon hole proof". That is, there are 29200 pigeon holes (dates), but there are 50000 pigeons(people). You can not put all the pigeons in a given number of holes. We have to reuse some of the holes. This proves the statement, "there must be at least two people who were born on the same day of the same year".
1. How many people do you need before you can be certain that two of them have the same number of hairs on their head?"
2. How many books will you need before you can be certain that two of them had exactly the same number of words?"
Statements like these can be proved using the pigeon hole principle.
PROOF BY CONTRADICTION:
Either multiplicand or multiplier must be greater than 8 to get the product 72. That is, in x*y = 72, either x or y must be greater than 8. Is it true?
If both are equal to 8, the product is 64 (8*8=64) - a contradiction. Hence one number must be greater than 8 to get 72. Another example: A member of parliament argues in these lines. "The Right Honourable gentleman claims that he will increase public spending. The only way in which he can achieve this is by increasing taxes -which he has already ruled out (a contradiction). I, therefore, pronounce his argument to be in tatters".
Hence proving is art also Math.
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