Home | 18.013A | Chapter 1 | Section 1.2 | | Tools Glossary Index Up Previous Next

Exercise 1.1

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 2 n + 1 to the positive integer n , and the integer 2 n 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.

Up Previous Next