# if $x\mathcal R y$ defined by $|x|+|y| =|x+y|$. Is it an equivalence relation?

Reflexive and symmetric can be proved as $|x|+|x|=|x+x|$ hence reflexive and $|y|+|x|=|y+x|$ hence symmetric but how transitive?

Hint: $|x|+|y|=|x+y|$ is true if $x$ and $y$ have the same sign or one of them is $0$.

Thus, all positive numbers are related to each other, and all negative numbers are related to each other, and $0$ is related to everything ...

• So it would be an equivalence relation on $\mathbb R \setminus \{0\}$. And it would be just a very uncommon way of describing the fibres of the sign function...
– MooS
Oct 28, 2015 at 20:29

It is not transitive. So no.

$1$ is related to $0$ but also $0$ is related to $-1$. Is it true that $1$ and $-1$ are related?

Hint: Question is not exactly defined. Relation can be equivalence, if you restrict the base set (from complex or real).