To actually construct the rationals ℚ from the integers ℤ, let S = {(a, b): a, b ∈ ℤ and b...

To actually construct the rationals ℚ from the integers ℤ, let S = {(a, b): a, b ∈ ℤ and b ≠ 0}. Define an equivalence relation on S by (a, b) ~ (c, d) iff ad = bc. Then define the set ℚ of rational numbers to be the set of equivalence classes corresponding to ~. Let the equivalence class determined by the ordered pair (a,b) be denoted by [a/b]. Then [a/b] is what we usually think of as the fraction a/b. For a, b, c, d ∈ ℤ with b ≠ 0 and d ≠ 0, we define addition and multiplication in ℚ by

[a/b]+[c/d]= [(ad + be)/bd],

[a/b] · [c/d] = [(ac)/(bd)].

We say that [a/b] is positive if ab ∈ ℕ. Since a, b ∈ ℕ. with b ≠ 0, this is equivalent to requiring ab > 0. The set of positive rationals is denoted by ℚ+, and we define an order “<” on ℚ by

x < y iff  y − x ∈ ℚ+.

(a) Verify that ~ is an equivalence relation on S.

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(b) Show that addition and multiplication are well-defined. That is, suppose [a/b] = [p/q] and [c/d] = [r/s]. Show that [(ad + bc)/bd] = [(ps + qr)/qs] and [ac/bd] = [pr/qs].

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(c) For any b ∈ ℤ \ {0}, show that [0/b] = [0/1] and [b/b] = [1/1].

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(d) For any a, b ∈ ℤ with b ≠ 0, show that [a/b]+ [0/1] = [a/b] and [a/b] − [1/1] = [a/b]. Thus [0/1] corresponds to zero and [1/1] corresponds to 1.

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(e) For any a, b ∈ ℤ with b ≠0, show that [a/b] + [(−a)/b] = [0/1] and [a/b] × [b/a] = [1/1].

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(f) Verify that the set ℚ with addition, multiplication, and order as given above satisfies the axioms of an ordered field.