Home  18.013A  Chapter 1  Section 1.2 


Is Z countable?
Solution:
Though there are "twice as many" positive and negative integers as there are only positive ones, we can make a onetoone 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.
