math1081-assignment/q2/d.md

Question 2

A relation \star is defined on the set \mathbb{Q}^{2} by \left(x_{1},x_{2}\right)\star\left(y_{1},y_{2}\right) if and only if x_{1}y_{2}=x_{2}y_{1}.

For each of the questions below, be sure to provide a proof supporting your answer.

Part D

Is \star transitive?

For \star to be transitive, then for any set of 3 elements (x_1,x_2), (y_1,y_2), and (z_1,z_2) where (x_1,x_2)\star(y_1,y_2) and (y_1,y_2)\star(z_1,z_2) are both true, then (x_1,x_2)\star(z_1,z_2) must also be true.

Consider a set of 3 elements (x_1,x_2), (y_1,y_2), and (z_1,z_2) where (x_1,x_2)\star(y_1,y_2) and (y_1,y_2)\star(z_1,z_2) are both true. This means that x_1y_2=x_2y_1 and y_1z_2=y_2z_1. From this, it can be found that y_1=\frac{y_2 z_1}{z_2}=\frac{x_1 y_2}{x_2}. This can be simplified to

\begin{aligned}\frac{z_1}{z_2}&=\frac{x_1}{x_2} \\ x_1&=\frac{z_1x_2}{z_2} \\ x_1z_2&=z_1x_2,\end{aligned}

which satisfies the requirement for transitivity. Therefore, the relation \star is transitive.