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Is Z countable?
Solution:
Though there are "twice as many" positive and negative integers as
there are only positive ones, we can make a one-to-one correspondence between
Z and N. We can, in other words, assign a unique positive integer to each positive
and negative integer.
How? Assign the positive integer 2n+1 to the positive integer n, and the integer
2n to the negative integer –n. The correspondence looks like this:
| N: |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
... |
| Z: |
0 |
-1 |
1 |
-2 |
2 |
-3 |
3 |
-4 |
4 |
-5 |
5 |
-6 |
6 |
-7 |
7 |
-8 |
... |
Sooner or later you get to every element of Z this way, though the elements
of N grow faster than those of Z. The peculiar fact, but fact nevertheless is
that it doesn't matter at all that the elements of N grow faster here.
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