I'm having difficulty with the following problem:
Prove: if $R$ is a weak partial (linear) order on $X$, then $R^− = R \; \backslash\ Id_X$ is a strict partial (linear) order.
I know that as a weak partial order, $R$ is reflexive, antisymmetric, and transitive. I know that the identity relation $Id_X$ is reflexive, asymmetric, symmetric/antisymmetric and transitive. I also know that as a strict partial order, $R^−$ is irreflexive, asymmetric, and transitive.
I can see that every reflexive relation in $R$ is also in $Id_X$, and that the difference $R \; \backslash\ Id_X$ will thus not contain any $<x, y>$ such that $x = y$, and will therefore be irreflexive.
I can see that this also implies that $R \; \backslash\ Id_X$ will be asymmetric, since the only $<x, y>$ pairs such that $x = y$ have been removed with $Id_X$ and by the definition of antisymmetry those were the only $<x, y>$ pairs for which $<y, x>$ was also the case.
I'm having trouble reasoning through why $R^−$ must be transitive. I think that if it weren't transitive, it would have to be the case that $R$ would also not be transitive, but I'm not sure how to articulate why that seems correct.
Finally, my biggest issue is just that I don't have math experience and have a hard time putting any of the above in formal notation.
Thanks for any help anyone is able to give.