Proving Equivalence of Operations on Equivalence Classes

[mossfan563]

Homework Statement

Prove that if (a1, b1) ~ (a2, b2) and (c1, d1) ~ (c2, d2), then (a1, b1) + (c1, d1) ~ (a2, b2) + (c2, d2)
and (a1, b1) \bullet (c1, d1) ~ (a2, b2)\bullet (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]\bullet [c, d] = [(a, b) (c, d)]; these definitions make sense, i.e., they do not depend on the choice of representatives.

Homework Equations

(a, b) + (c, d) = (ad + bc, bd) and (a, b) \bullet (c, d) = (ac, bd)
(a, b) ~ (c, d) if and only if ad = bc

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?

 

[Billy Bob]

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

Are you asking how to prove this?

Write down what (a1, b1) ~ (a2, b2) means and what (c1, d1) ~ (c2, d2) means; these are given.

Write down what (a1, b1) + (c1, d1) is, what (a2, b2) + (c2, d2) is, and then what (a1, b1) + (c1, d1) ~ (a2, b2) + (c2, d2) means. This is what you must prove. The proof practically writes itself once you write down what these statements mean.

Repeat for *, which may require a tiny trick.

Write out your work for us and ask questions if you get stuck.