The confusions stem from this question:
Let $R$ be a relation over a set $A$. For all $a∈A$ and $b∈A$, given that $a < b$ iff $a\leq b$ but $a\neq b$, and $a\leq b$ iff either $a < b$ or $a=b$, show that when $<$ is a transitive asymmetric relation, then the relation $\leq$ is a partial order.
I can prove that $a\leq b$ is transitive and anti-symmetric, but I am not sure how to prove that $a \leq b$ is reflexive.
My focus is on proving $a=b$, because trying to prove $a < b$ seems impossible. My plan was to argue that since $a$ and $b$ are just arbitrary objects from set $A$, $a$ and $b$ could be referring to the same object. If so, necessarily $a=b$. But is this a valid move? The fact that one can simply assume two variables to be the same thing seems like an 'illegal' move, in the sense that it grants one a very powerful assumption.
My next confusion concerns whether an inequality is symmetric. This is because it could make my anti-symmetric proof shorter. It is well known that equality is symmetric, but the same is not apparent for inequality - at least as far as Google can tell me. Could anyone tell me if that is true?