Friday, September 06, 2002

Experience Mathematics #12 -- A part can be equal to the whole!

A set S is countable if it can be put in one-to-one correspondence with $N$. For example, if we take the set of even numbers, we can establish a one-to-one correspondence as follows. $1$ corresponds to $2$, $2$ to $4$, $3$ to $6$, and so on. This shows that the number of even numbers is equal to the number of natural numbers.

This contradictory idea—that a part of an object is equal to the whole—troubled many philosophers. But they got over it, and began to compare the concept of infinity with the concept of God. However, when Biologists made it possible to clone human beings, they have stopped approving of the idea.

The famous mathematician Hilbert told the story of a hotel with an infinite number of rooms. Suppose the Hilbert Hotel is full, but the hotel manager wants to accommodate a guest who arrives suddenly. How does he manage that? Well, he asks the guest in room number $1$ to move to Room $2$, the guest in Room $2$ to move to Room $3$, and so on. Room $1$ becomes empty and is readied for the new guest. Can you figure out how to accommodate $30$ guests, even if the hotel is full? 

What if a travel agent calls the manager, and says she is sending groups of tourists to the hotel. The numbers of people in the groups are: $3$, $7$, $11$, $15$, and so on. Help the manager come up with the required one-to-one correspondence in these cases, so that he can accommodate all the guests.

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