An ordering relation $\precsim$ on $X$ is said to be preordered if it satisfies the following two conditions:
(i) reflexive: $x \precsim x$ for all $x \in X$,
(ii) transitive: if $x \precsim y$ and $y \precsim z$, then $x \precsim z$.
Give an example of a preoder on $\mathbb{R}$ which is not a partial order.
I have tried $x \precsim y$ if and only if $x \leq 2y$. Also, I have tried $x \precsim y$ if and only if $x \leq \frac{y}{2}$. In the first one transitivity is not satisfied and in the second one reflexivity is not satisfied.
0 < 1, 1 < 0 and for all x, x < x.