Let $\preceq$ be a reflexive relation on a set $X$ and let $\prec$ be the corresponding irreflexive relation. Then
(i) $\preceq$ is symmetric if and only if $\prec$ is symmetric, and
(ii) $\preceq$ is transitive if and only if $\prec$ is transitive.
It's easy exercise, but i can't see why necessity in (ii) is true?
(ii) is false. If $X=\{a,b\}$ and $\preceq$ satisfies $x\preceq y$ for all $x$ and $y$, then $\preceq$ is transitive but $a\prec b\prec a$ and $a\not\prec a$ so that $\prec $ is not transitive.