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(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.

Surdz
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  • What difficulties did you face? What did you try to do to prove that the relation is transitive, and where did it go wrong? – qaphla May 19 '16 at 03:59
  • well I just want to show that is transitive: aRb and bRc then aRc, if a is related to be and b is related to c then a related to c. I started ad=bc and bc=? thats where I have trouble. – Surdz May 19 '16 at 04:05

1 Answers1

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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.

Brandon
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