(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that if (a1, b1) ~ (a2, b2) and (c1, d1) ~ (c2, d2), then (a1, b1) + (c1, d1) ~ (a2, b2) + (c2, d2)

and (a1, b1) [tex]\bullet[/tex] (c1, d1) ~ (a2, b2)[tex]\bullet[/tex] (c2, d2).

Let [a, b] denote the equivalence class with respect to ~ of (a, b) in Z x (Z-{0}), and define Q to be the set of equivalence classes of Z x (Z-{0}).

For all [a, b], [c, d] in Q define [a, b] + [c, d] = [(a, b) + (c, d)] and [a, b][tex]\bullet[/tex] [c, d] = [(a, b) (c, d)]; these definitions make sense, i.e., they do not depend on the choice of representatives.

2. Relevant equations

(a, b) + (c, d) = (ad + bc, bd) and (a, b) [tex]\bullet[/tex] (c, d) = (ac, bd)

(a, b) ~ (c, d) if and only if ad = bc

3. The attempt at a solution

I tried using those definitions.

I know you have to assume that (a1, b1) ~ (a2, b2) and (c1, d1) ~ (c2, d2).

But I get stuck afterwards. Where do I go from there? Do I need something more?

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# Equivalence class proof

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