I found this question and answer on a model paper. However I believe the answer is wrong. Can you help me understand please?
Question: 1. Consider the relation R on the set of real numbers R defined by xRy if and only if xy is a rational number. i. Symmetric ii. Reflexive iii. Transitive
Answers given i. As xy = yx for all, x,y ∈ R we have the yx is rational if xy is rational, and so R is symmetric. ii. R is not reflexive as e.g. pi*pi is not rational. iii. R is not transitive as e.g. pi/2 * 2/pi = 1 is rational, while pi/2*pi/2 is irrational.
My concerns: a. I think answer (i) is wrong. Here is why... x = square root of 5 and y = square root of 2, xy = square root of 10 which is irrational. Not for all x,y we have yx rational. Is the model paper answer wrong here?
I think (ii) and (iii) are correct. Right?
Thank you very much!
Your example does not prove i is wrong. If $x=\sqrt 5, y=\sqrt 2, xy=yx=\sqrt {10}$ then both $xy$ and $yx$ are irrational, so we have $x \not R y$ and $y \not R x$. The iff condition is satisfied because both statements $xRy$ and $yRx$ are false. The point is that because multiplication is commutative we have that either both $xy$ and $yx$ are rational or both are irrational.