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The equivalence classes

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Reference no: EM13334438

Consider the relation on X = Z × N + given by (a, b) ≈ (c, d) if and only if ad = bc. (As proved in the homework, this is an equivalence relation.) Consider the operation
∗ : X × X → X (x, y) ∗ (z, w) = (xw + yz, yw).
Show that this gives a well-defined map ∗ : X/≈×X/≈ → X/≈ on the equivalence classes.
(Warning: For full credit, your proof should not make use of the rational numbers. This means that (assuming x, y, z ∈ Z) you may use the fact that if xy = xz and x 6= 0 then y = z, but your solution should avoid arithmetic with rational numbers.)

Reference no: EM13334438

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