Question

Let R be a relation on the set A of ordered pair of integers defined by (x, y) R (u, v) if xv = yu. Show that R is an equivalence relation.

Answer 1

We observe the following properties of R.

$$\begin{gathered} \mathrm{Reflexivity}:\mathrm{Let}\left(a,b\right)\mathrm{be}\mathrm{an}\mathrm{arbitrary}\mathrm{element}\mathrm{of}\mathrm{the}\mathrm{set}\mathrm{A}.\mathrm{Then}, \\ \left(a,b\right)\in A \\ \Rightarrow ab=ba \\ \Rightarrow \left(a,b\right)R\left(a,b\right) \\ \mathrm{Thus},R\mathrm{is}\mathrm{reflexive}\mathrm{on}A. \\ \mathrm{Symmetry}:\mathrm{Let}\left(x,y\right)\mathrm{and}\left(u,v\right)\in A\mathrm{such}\mathrm{that}\left(x,y\right)R\left(u,v\right).\mathrm{Then}, \\ xv=yu \\ \Rightarrow vx=uy \\ \Rightarrow uy=vx \\ \Rightarrow \left(u,v\right)R\left(x,y\right) \\ \mathrm{So},R\mathrm{is}\mathrm{symmetric}\mathrm{on}A. \\ \mathrm{Transitivity}:\mathrm{Let}\left(x,y\right),\left(u,v\right)\mathrm{and}\left(p,q\right)\in R\mathrm{such}\mathrm{that}\left(x,y\right)R\left(u,v\right)\mathrm{and}\left(u,v\right)R\left(p,q\right). \\ \Rightarrow xv=yu\mathrm{and}uq=vp \\ \mathrm{Multiplying}\mathrm{the}\mathrm{corresponding}\mathrm{sides},\mathrm{we}\mathrm{get} \\ xv\times uq=yu\times vp \\ \Rightarrow xq=yp \\ \Rightarrow \left(x,y\right)R\left(p,q\right) \\ \mathrm{So},R\mathrm{is}\mathrm{transitive}\mathrm{on}A. \end{gathered}$$

Hence, R is an equivalence relation on A.