Let R be the relation on the set of ordered pairs of positive integers, Z+ × Z+, such that (a, b)R(c, d) if and only if ad = bc.

Question

(For instance, (2, 4)R(6, 12) since 2·12 = 4·6.) Show that R is an equivalence relation.

I was tasked to show that the sets is an equivalence relation if the three conditions Reflexive, symmetric and transitive is shown valid, However I faced difficulties when showing transitive?

I really need hints. Thanks.

Answer 1

Assume that $(a,b)R(c,d)$ and $(c,d)R(e,f).$ We see that this implies that $$ad = bc \wedge cf = de.$$ We want to show that $$af = be.$$ Given that $b \neq 0.$ We see that $$\frac{ad}{b} = c \implies \frac{adf}{b} = de.$$ $$\implies adf = bde.$$ Given taht $d \neq 0$, We see that $$af = be \implies (a,b)R(e,f).$$ And we have thus proved transitivity.