Set A = Z times (Z {0}) = {(x, y) Z times Z | y 0} and define a relation R on A

Question

Set A = Z times (Z {0}) = {(x, y) Z times Z | y 0} and define a relation R on A by (x, y)R(u, v) : xv = yu. Show that R is an equivalence relation. What is the "meaning" of this relation and what can you identify the quotient set with?

Explanation / Answer

(Reflexive) xy = xy = yx so (x,y)R(x,y) (Symmetric) Let (a,b)R(c,d). Then ad = bc => cb = da. Thus (c,d)R(a,b) (Transitive) Let (a,b)R(c,d) and (c,d)R(e,f). Then ad = bc and cf = de. Since b and f are nonzero, then c = ad/b and c = de/f => ad/b = de/f Since d is nonzero, then a/b = e/f => af = be. Therefore (a,b)R(e,f) and R is an equivalence relation. Notice that (a,b) is similar to fractions you are familiar with having a as the numerator and b as the denominator. So two things are related if they "reduce" to the same fraction. That is, since 1/2 = 2/4, you also have (1,2)R(2,4) since 1*4 = 2*2 = 8. So quotient sets are the ones with that reduce to the same fraction.

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