math1081-assignment/q2/a.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 A

Is \star reflexive?

A relation on \mathbb{Q}^2 is reflexive if a\star a for all a\in\mathbb{Q}^2. To prove reflexivity, assume that x_{1}x_{2}=y_{2}y_{1} is true (let this be (1)).

Given that x_1 y_2 = x_2 y_1 is known to be true, we can rearrange for y_1 to find that y_1=\frac{x_1 y_2}{x_2} (2). In a similar manner, rearranging for y_2 provides y_2=\frac{x_2 y_1}{x_1} (3). Substituting (2) and (3) into (1), it can be found that x_2=y_2 and x_1 = y_1. Therefore, x_{1}x_{2}=y_{2}y_{1}=x_1 y_2 = x_2 y_1. Since every (x_1,x_2) satisfies the requirement that x_{1}x_{2}=y_{2}y_{1}, the relation is reflexive.