Examples 2.1.7(b):

The set of all polynomials that have integer coefficients and degree n is countable.
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Let P(n) be the set of all polynomials with integer coefficients and degree n. Then a particular element of P(n) is

p_n(x) = a_nxⁿ + a_{n - 1}x^{n - 1} + a_{n - 2}x^{n - 2} + ... + a₁x + a₀

Define a function f as follows:

domain of f is P(n), range of f is Z x Z x ... x Z (n+1 times)
f(p_n) = f( a_nxⁿ + a_{n - 1}x^{n - 1} + ... a₁x + a₀) = (a_n, a_{n - 1}, ..., a₁, a₀)

Because 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 that P(n) is also countable.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007