## 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.