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(1) (a) describe why Z X Z must be countable.

(b) By part (a) we know that N ≡ Z X Z, and hence there must be a one-to-one correspondence between N and Z X Z. Provide one such one-to-one correspondence f : N → Z X Z. Note, it may be easiest to describe your map using a sketch of Z X  Z.

(2) Proved in that a finite product of countable sets is countable. It is not true however that a countable product of count- able sets must be countable. Here you will see one ex of a countable product of  finite sets that is not countable.

For each i ≡ N let Ai = {0,1} and let

1150_Finite sets.jpg

the set of all sequences of 1's and 0's. The set A is a countable product of  finite and hence countable sets. Prove that A is not countable.
(3) A complex number z is "algebraic" if it is a root of some integral polynomial (a polynomial with integer coefficients). That is z is algebraic if there is some

p(x) = a0 + a1x + a2x2 +........+ akxk ai ≡ Z and at least one ai ≠ 0 with p(z) = 0. For ex, the polynomial p(x) = -4+4x-x2+x3 factors to p(x) = (x- 1)(x - 2i)(x + 2i) and so has roots x = 1 and x = ± 2i, thus 1, 2i and -2i are algebraic numbers. Let A denote the set of all algebraic numbers in C, and prove that A is countable by the following steps.

(a) For each n ≡ N, let Pn denote the set of integral polynomials of degree n. Prove that for each n the set Pn is countable.

(b) Now let P be the set of all integral polynomials. describe why P must be countable.

(c) Given a particular integral polynomial p(x) of degree k, let Rp be the set of all roots of p(x). What can be said about the number of elements in Rp?

(d) Using parts (a), (b) and (c), prove that A, the set of all algebraic integers, is countable.

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