Completeness of the strict binary relation How do I show that considering the preference relation $\succsim$, then  $\succ$ is not complete?

I tried the following (which I don't know if it's right) but I'd also like to know if it's possible to generalize and say that $\succ$ will never be complete.

Suppose $X = \{a,b\}$ with preference relation $\{ a \succsim a, a \succsim b, b \succsim b\}$. Then, we $\succ$ can't be a complete relation on $X$ because neither $b \succsim a$ nor $b \not{\succsim} a$ are in $X$ and therefore it's impossible to conclude that either $a \succ b$ or $b \succ a$ is true.
Your counterexample looks correct.

But an even simpler example would be $X=\{a\}$. We have $a \succsim a$, so $a \not \succ a$.



More generally, we usually define


$\succsim$ to be reflexive (i.e. $a \succsim a$ for all $a$); and
$\succ$ by $a \succ b$ if $a \succsim b$ and $b \not \succsim a$.


By the above definitions, $\succ$ cannot be complete on any non-empty set $X$ because for any $a\in X$, we have $a \not \succ a$.