My book has an example that goes like this:
$A = \{1,2,3,4\}$ $R = (G,A,A)$ Prove that $R$ is antisymmetric if and only if $G \cap G^{-1} \subseteq D$
We have to prove two implications. The first one being "supposing that $R$ is antisymmetric, prove that $G \cap G^{-1} \subseteq D$ occurs". So here it goes:
$(a,b) \in G \cap G^{-1} \implies (a,b) \in G \land (a,b) \in G^{-1}$ $\implies aRb \land aR^{-1}b \implies aRb \land bRa$
Since $aRb \land bRa$, using our hypothesis that $R$ is antisymmetric, we can say that $a = b$. Therefore, $(a,b) \in D$
There are two things I don't entirely understand in this proof:
- What is $D$? Is it an arbitrary set or something else I'm missing?
- How exactly can proving that $a = b$ imply that $(a,b) \in D$?