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Jim Lambers Math 6A Spring Quarter 2003- Lecture 21 Examples
These examples correspond to Section 3.2 in the text.
Example The set of integers, Z, is countable. To see this, we note that a bijection between N and Z can be explicitly constructed. The function f : Z โ N defined by
f (z) =
2 z if z โฅ 0 โ 2 z โ 1 if z < 0 ,^ z^ โ^ Z,
has the inverse f โ^1 : N โ Z given by
f โ^1 (n) =
n/ 2 if n is even โ(n + 1)/ 2 if n is odd
That these functions are inverses can be seen by computing f (f โ^1 (n)) and f โ^1 (f (z)) and verifying that they are equal to n and z, respectively. 2
Example The set of rational numbers, Q, is countable. This can be seen by writing the positive rational numbers on an infinite grid, starting at the upper left corner and writing numbers on diagonals further and further away from the corner, in such a way that the numbers on each diagonal have the same sum of their numerator and denominator. This grid begins as follows:
1 / 1 2 / 1 3 / 1 4 / 1 5 / 1 6 / 1 1 / 2 2 / 2 3 / 2 4 / 2 5 / 2 1 / 3 2 / 3 3 / 3 4 / 3 1 / 4 2 / 4 3 / 4 1 / 5 2 / 5 1 / 6
By writing the positive rational numbers in this order, we can be sure to list all of them. If we associate positive integers with these rational numbers as follows:
1 / 1 โ 1 2 / 1 โ 3 3 / 1 โ 6 4 / 1 โ 10 5 / 1 โ 15 6 / 1 โ 21 1 / 2 โ 2 2 / 2 โ 5 3 / 2 โ 9 4 / 2 โ 14 5 / 2 โ 20 1 / 3 โ 4 2 / 3 โ 8 3 / 3 โ 13 4 / 3 โ 19 1 / 4 โ 7 2 / 4 โ 12 3 / 4 โ 18 1 / 5 โ 11 2 / 5 โ 17 1 / 6 โ 16
then we can describe an explicit bijection between Q and Z+, the set of positive integers. We can define f : Q โ Z+^ by
f (p/q) = (p + q โ 1)(p + q โ 2) 2
while its inverse f โ^1 : Z+^ โ Q is defined by
f โ^1 (n) = p q
, k = p + q =
1 + 8n 2
, p = n โ (k โ 1)(k โ 2) 2
, q = k โ p.
These functions can be obtained by noting that first diagonal has one number, the second has two, and so on, with the jth diagonal containing j numbers. Therefore the number of rational numbers on the first j diagonals is
โj i=1 i^ =^ j(j^ + 1)/2. Also note that the sum of the numerator and denominator is constant along each diagonal, and is equal to j +1, where j is the number of the diagonal. The fact that n, the positive integer associated with each rational number, is related to p + q, the sum of the numerator and the denominator, by a quadratic function in the definition of f implies that the quadratic formula must be used in obtaining f โ^1. Given that the positive rational numbers are countable, it can be shown that the set of all rational numbers, including zero and negative rational numbers, is also countable, in the same way that the integers are countable.
