## 2.1. Countable Infinity

### Examples 2.1.5:

The set of integers Z and the interval of real numbers between 0 and 2, [0, 2],
are both infinite.

According to Dedekind's theorem, we need to find a proper subset of each
set that has the same cardinality as the original set. In other words, we
need to find a bijection from the original set into a proper subset of
itself.
Define the function *f(n) = 2n*. Then this function is a bijection
between **Z** and the even integers. Hence, **Z** has the same
cardinality as a proper subset of itself, and therefore **Z** is infinite.

Define the function *f(x) = x / 2*. Then the function is a bijection
between the interval [0, 2] and the interval [0, 1]. Hence, the interval
[0, 2] is infinite.