Question

Let $\mathit{S}\mathbf{=}\mathbf{\{}\left(a,b\right)\overline{)\mathbf{}\mathbf{a}\mathbf{,}\mathbf{b}}\mathbf{\in }\mathbf{\square }\mathbf{\square }\mathbf{}\mathbf{a}\mathbf{n}\mathbf{d}\mathbf{}\mathbf{b}\mathbf{\ne }\mathbf{0}\mathbf{\}}$ and define$\left(a,b\right)\mathbf{\square }\left(c,d\right)$ if and only if $\mathit{a}\mathit{d}\mathbf{=}\mathit{b}\mathit{c}$. Prove that$\mathbf{\square }$ is an equivalence relation on S.

Short answer

We proved that$\square$ is an equivalence relation.

Step-by-step solution

To mention given data

Let $S=\{(\mathrm{a},\mathrm{b})\overline{)\mathrm{a},\mathrm{b}}\in \square \mathrm{and}\mathrm{b}\ne 0\}$and define $(a,b)\square \left(c,d\right)$if and only if ad = bc.

We have to prove that$\square$ is an equivalence relation.

In order to prove that$\square$ is an equivalence relation, we need to prove the following conditions:

1. ‘$\square$’ is reflexive.
2. ‘$\square$’ is symmetric.
3. ‘$\square$’ is transitive.

To prove that □ is reflexive

Let $a,b\in \square$.

Then:

$$\begin{gathered} \left(a,b\right)\square \left(a,b\right) \\ \iff ab=ba \end{gathered}$$

Hence,$\square$ is reflexive.

To prove that □ is symmetric

Let $a,b,c,d\in \square$.

Then suppose that:

$$\begin{gathered} \left(a,b\right)\square \left(c,d\right) \\ \iff ad=bc \\ \iff cb=da \\ \iff \left(c,d\right)\square \left(a,b\right) \end{gathered}$$

Hence,$\square$ is symmetric.

To prove that □ is transitive

Let $a,b,c,d,e,f\in \square$.

Then suppose that:

$$\begin{gathered} \left(a,b\right)\square \left(c,d\right),\left(c,d\right)\square \left(e,f\right) \\ \iff ad=bc,cf=de \\ \iff adcf=bcde \\ \iff af=be \\ \iff \left(a,b\right)\square \left(e,f\right) \end{gathered}$$

Hence,$\square$ is transitive.

Hence,$\square$ is an equivalence relation since all the conditions are satisfied.