I have a relation $R$ on an arbitrary set $X$ not necessarily containing numbers, $R$ is transitive, asymmetric and incomplete, I am to show that for any such relation $R$ we can find a set of transitive,asymmetric AND complete relation $R_i$ such that for any $a,b$ in $X$ we have $aRb$ iff $aR_ib$ for all $i$.
Example $X = (a,b,c)$ and say that we have $aRb$ but $b$ and $c$ cant be compared and $a$ and $c$ can’t be compared either. So given this I construct the following
$a R_1bR_1c$
$aR_2cR_2b$
So these two give me the required $R$ on $X$ but I can’t figure out how to generalize this to any arbitrary $X$ having some pairs that are comparable and others that are not.