A B C 
Homework

Problem H2

If we think of division as repeated subtraction, in essence, we are subtracting groups of 0 from the number we are dividing. It is easy to see that you could subtract groups of 0 infinitely many times and never exhaust the number you started with. Therefore, dividing by 0 is not defined.

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Problem H3

The easiest solution would be to move everyone in the hotel one room over. This way, the first room would be freed up for the traveler. Notice that here you've added one element to an infinitely countable set. The result is still an infinitely countable set:

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Problem H4

To put everyone into a room, you would alternate the guests (G) already there and the marching band guests (MB). In other words, the guests who were already in the hotel would be moved into rooms with odd numbers:

By combining the two infinite sets in this way, you still get a countably infinite set that can be put into one-to-one correspondence with the counting numbers.

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Problem H5

By writing down all the rooms in all the hotels in an infinite two-dimensional matrix (see below), they can all then be put into one-to-one correspondence with the counting numbers, each of which corresponds to a particular room:

Notice that here you have multiples of infinitely countable sets combined into one set. The new set is also countably infinite, which you've shown by putting it into one-to-one correspondence with the counting numbers.

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