$A$ is the "absolute value" relation on $\Bbb R$: For all real numbers $x$ and $y$, $x A y\Leftarrow\Rightarrow |x| = |y|$.
Determine whether the given relation is reflexive, symmetric, or transitive, or none of these. Justify your answer.
I am struggling with the proofs for this question. I believe it is all three, but I have confused myself overthinking it.
To show reflexivity, note that for every $x\in R$, we have $xAx$ (just by definition)
To show symmetry, note that for every $x,y\in R$, we have $xAy$ implies $yAx$(again just by definition)
To show transitivity, note that for every $x,y,z\in R$, we have $xAy$ and $yAz$ implies $xAz$ because $xAy$ means $\vert x \vert =\vert y \vert$ and $yAz$ means $\vert y \vert =\vert z \vert$ therefore $\vert x \vert =\vert z \vert$,hence $xAz$.
