math1081-assignment/q2/c.md

Part C

Is \star anti-symmetric?

If \star is anti-symmetric, then for every distinct pair of \left(x_{1},x_{2}\right) and \left(y_{1},y_{2}\right) in \Q^2 such that \left(x_{1},x_{2}\right)\star\left(y_{1},y_{2}\right) and \left(y_{1},y_{2}\right)\star\left(x_{1},x_{2}\right), x_1=y_1 and x_2=y_2.

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, the relation is anti-symmetric.