make statements of properties of relations uniform; closes #226

content/sets-functions-relations/relations/special-properties.tex

@@ -67,13 +67,19 @@
 \end{prob} 
 
 \begin{defn}[Irreflexivity]
-A relation $R$ on $A$ is called \emph{irreflexive} if, for all $x \in
-A$, $ \lnot Rxx$. 
+A relation $R \subseteq A^2$ is called \emph{irreflexive} if, for all $x \in
+A$, not $Rxx$. 
 \end{defn}
 
 \begin{defn}[Asymmetry]
-A relation $R$ on $A$ is called \emph{asymmetric} if for no pair $x,y\in
-A$ we have $Rxy$ and $Ryx$. 
+A relation $R \subseteq A^2$ is called \emph{asymmetric} if for no pair $x,y\in
+A$ we have both $Rxy$ and~$Ryx$. 
 \end{defn}
 
+Note that if $A \neq \emptyset$, then no irreflexive relation on~$A$
+is reflexive and every asymmetric relation on~$A$ is also
+anti-symmetric. However, there are $R \subseteq A^2$ that are not
+reflexive and also not irreflexive, and there are anti-symmetric
+relations that are not asymmetric. 
+
 \end{document}