Is this an equivalence relation? $xRy$ if and only if $|x+y| = |x| + |y|$

Question

I can prove that it is reflexive, but I am not sure how to formally prove that this relation is symmetric and transitive.

Answer 1

Hint:

$$|1+0|=|1|+|0|$$

$$|0-1|=|0|+|-1|$$

To check symmetric:

Notice that $|x+y|=|y+x|$ and $|x|+|y|=|y|+|x|$, hence if $|x+y|=|x|+|y|$, what can you conclude about $|y+x|$.

Answer 2

Pick, as in the above hint, $x=-1$ $y=0$ $z=2$

We have that $xRy$, because $|x+y|=|-1 + 0| = |-1| + |0|=|x|+|y|$,

$yRz$, because $|y+z| = |0+2| = |0| + |2|$

But from the fact that the following is false $1=|-1+2| = |-1| + |2|=3$ it does not follow that $xRz$.