]> Exercise 1.1
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## 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.

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