Examples 2.1.7(b):
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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
pn(x) =
anxn +
an - 1xn - 1 +
an - 2xn - 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(pn) = f(
anxn +
an - 1xn - 1 + ...
a1x +
a0)
=
(an, an - 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.