Question

Let Z be the set of all integers and Z₀ be the set of all non-zero integers. Let a relation R on Z × Z₀ be defined as
(a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z₀,
Prove that R is an equivalence relation on Z × Z₀.

Answer 1

We observe the following properties of R.

Reflexivity:
$$\begin{gathered} \mathrm{Let}\left(a,b\right)\mathrm{be}\mathrm{an}\mathrm{arbitrary}\mathrm{element}\mathrm{of}\mathrm{Z}\times {\mathrm{Z}}_0.\mathrm{Then}, \\ \left(a,b\right)\in \mathrm{Z}\times {\mathrm{Z}}_0 \\ \Rightarrow a,b\in \mathrm{Z},{\mathrm{Z}}_0 \\ \Rightarrow ab=ba \\ \Rightarrow \left(a,b\right)\in R\mathrm{for}\mathrm{all}\left(a,b\right)\in Z\times Z_0 \\ \mathrm{So},R\mathrm{is}\mathrm{reflexive}\mathrm{on}Z\times Z_0. \end{gathered}$$

Symmetry:
$$\begin{gathered} \mathrm{Let}\left(a,b\right),\left(c,d\right)\in Z\times Z_0\mathrm{such}\mathrm{that}\left(a,b\right)R\left(c,d\right).\mathrm{Then}, \\ \left(a,b\right)R\left(c,d\right) \\ \Rightarrow ad=bc \\ \Rightarrow cb=da \\ \Rightarrow \left(c,d\right)R\left(a,b\right) \\ \mathrm{Thus}, \\ \left(a,b\right)R\left(c,d\right)\Rightarrow \left(c,d\right)R\left(a,b\right)\mathrm{for}\mathrm{all}\left(a,b\right),\left(c,d\right)\in Z\times Z_0 \\ \mathrm{So},R\mathrm{is}\mathrm{symmetric}\mathrm{on}Z\times Z_0. \end{gathered}$$

Transitivity:
$$\begin{gathered} \mathrm{Let}\left(a,b\right),\left(c,d\right),\left(e,f\right)\in N\times N_0\mathrm{such}\mathrm{that}\left(a,b\right)R\left(c,d\right)\mathrm{and}\left(c,d\right)R\left(e,f\right).\mathrm{Then}, \\ \left.\begin{array}{r}\left(a,b\right)R\left(c,d\right)\Rightarrow ad=bc\\ \left(c,d\right)R\left(e,f\right)\Rightarrow cf=de\end{array}\right\}\Rightarrow \left(ad\right)\left(cf\right)=\left(bc\right)\left(de\right) \\ \Rightarrow af=be \\ \Rightarrow \left(a,b\right)R\left(e,f\right) \\ \mathrm{Thus}, \\ \left(a,b\right)R\left(c,d\right)and\left(c,d\right)R\left(e,f\right)\Rightarrow \left(a,b\right)R\left(e,f\right) \\ \Rightarrow \left(a,b\right)R\left(e,f\right)\mathrm{for}\mathrm{all}\mathrm{values}\left(a,b\right),\left(c,d\right),\left(e,f\right)\in N\times N_0 \\ \mathrm{So},R\mathrm{is}\mathrm{transitive}\mathrm{on}N\times N_0. \end{gathered}$$