Let $f \colon \mathbb{R} \to \mathbb{R}$ be the well-known absolute value function: $f(x) = \lvert x \rvert$. Show that f, considered as a relation, is neither reflexive nor symmetric. Show also that it is transitive.
I'm not sure how to proceed on this one. Is the absolute value function not an equivalence relation?
This relation is defined by
$$(\forall x,y\in\Bbb R)\; (x \mathcal R y\iff y=|x|)$$
So, it is not reflexive since $-1$ is not in relation with $ -1 $ because $|-1|\ne -1$.
It is not symetric since $-1 $ is in relation with $ 1$ but $ 1 $ is not.
It is transitive :
$$x\mathcal Ry \quad \text{ and } \quad y\mathcal R z\implies$$ $$|x|=y \quad \text{ and } \quad|y|=z \implies |x|=|y|=z$$
$$\implies x \mathcal R z$$