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How to show that there are as many natural numbers as there are integers?
Hi, I'm just a little confused.How can I set up a one to one correspondence between the integers and the natural number to show that there are as many natural numbers as there are integers. And similarly how can I show that there is no one-to-one correspondence between the irrational numbers and the natural numbers.
Any help would be appreciated. Thank you !!!!
First, think of it this way:
Let 1, 2..... be the counting (natural numbers)
then 1 ==> 0
2 ==>1
3==>-1
4==>2
5==>-2
6==>3
etc.
As you can see, all the even numbers in the natural numbers are mapped onto a positive integer and all the odd natural numbers (except 1) are mapped onto negative integers. Since we never run out of natural numbers, we can "count" all the integers. the same thing can be done with the rational numbers.
to show there's no 1-1 correspondence between "irrational numbers" and the natural numbers, suppose that there is such a mapping. Then, if a and b are irrational numbers that are mapped, there must be some c, such that a