Is the reflexive closure of an asymmetric relation an antisymmetric relation?

Question

An asymmetric relation is a binary relation $R$ that satisfies the sentence $xRy \rightarrow \neg yRx$, for all $x$ and $y$. An antisymmetric relation satisfies the sentence $(xRy \land yRx) \rightarrow x=y$, for all $x$ and $y$. I wonder, is the reflexive closure of an asymmetric relation an antisymmetric relation?

Answer 1

Yes. Let $R$ be an asymmetric relation on $A$. Denote by $R'=R \cup \Delta$ the reflexive closure of $R$, where $\Delta=\{(a,a)\mid a\in A\}$. Take $x,y$ satisfying the sentence $xR'y \land yR'x$. If $x\neq y$, then since $(x,y)\not\in \Delta$, we state that $x$ and $y$ satisfy $xRy \land yRx$, which is a contradiction. So $x=y$ and $R'$ is antisymmetric.