Zodd as follows: let h(x)
= 2x + 1 for all integers x.1. Let Zodd represent the set of odd integers, and define a
function h: ZZodd as follows: let h(x)
= 2x + 1 for all integers x.
The function h is one-to-one.
Proof Suppose that there are two integers x1 and x2 such that h(x1) = h(x2). Then 2x1 + 1 = 2x2 + 1 and hence x1 = x2.
The function h is onto.
Proof Suppose there is an odd integer y. Then there exists an integer x such that h(x) = y. Here x = (y - 1)/2 which is an integer since y is odd and therefore y-1 is divisible by 2.
Since the function h is one-to-one and onto it is a one-to-one correspondence. Hence Zodd has the same cardinality as Z. It was shown in the textbook that the set Z of all integers is countable. Therefore the set of all odd integers is also countable.