## 2.1. Countable Infinity

### Examples 2.1.7(b):

Let**P(n)**be the set of all polynomials with integer coefficients and degree

*n*. Then a particular element of

**P(n)**is

Define a functionp_{n}(x) = a_{n}x^{n}+ a_{n - 1}x^{n - 1}+ a_{n - 2}x^{n - 2}+ ... + a_{1}x + a_{0}

*f*as follows:

domain ofBecause all coefficients are integers, this functions is onto, and is clearly one-to-one. Hence it is a bijection between the domain and the range. But because the finite cross product of countable sets is countable, this implies thatfisP(n), range offisZxZx...xZ(n+1times)

f(p_{n}) = f( a_{n}x^{n}+ a_{n - 1}x^{n - 1}+ ... a_{1}x + a_{0}) = (a_{n}, a_{n - 1}, ..., a_{1}, a_{0})

**P(n)**is also countable.