Having trouble with symmetry (equivalence relation)

Question

Define a relation of $x,y \in R$ when $x = |y|$.

I know this is reflexive as $x = |x|$ holds true because the relation has to have x as positive since $x = |y|$ which makes $x$ have to be positive or $0$ to be in the defined relation.

Transitivity also follows...

But for symmetery I have a question:

Would it be $|y| = x$ or rather would it be: $y = |x|$ ?

Thanks.

Answer 1

If the relation is defined on the real numbers then it isn't reflexive since for example $\;-1\rlap{\,/}R\, -1\;$, because $\;-1\neq|-1|=1\;$

It isn't symmetric since $\;1R-1\;$ but $\;-1\rlap{\,/}R\;1\;$

I'll leave transitivity for you, which shall be easier now after the above.

Answer 2

For symmetry, you need $xRy \implies yRx$ Using the definition of $R$ that would mean that $x=|y| \implies y=|x|$. Is that true?