1
$\begingroup$

The first definition of antisymmetric of a relation is $(a,b) \in R$ and $(b,a) \in R \implies a=b$. There is another saying if $a \neq b$ and $(a,b)\in R$, then $(b,a) \not\in R$. I could intuitively understand this or even by truth table, but I want to show how to transform one into another. I had gone with contrapositive and logical equivalence, neither work out. Here is what I got so far: $a\neq b \implies (a,b)\not\in R $ or $(b,a)\not\in R$ I cannot go further with this one.

Could someone show me how to do it, please?

$\endgroup$
1
$\begingroup$

Let $p$ be $(a,b)\in R$, $q$ be $(b,a)\in R$, and $r$ be $a=b$.

$(p\wedge q)\to r$ is equivalent to $p\to( q\to r)$ is equivalent to $p\to( \neg r\to \neg q)$ is equivalent to $(p\wedge \neg r)\to \neg q$.

$\endgroup$
  • 1
    $\begingroup$ Nice !! Thank you so much! Exactly what I am looking for $\endgroup$ – Lawrence Guo Aug 24 at 3:07
  • $\begingroup$ @LawrenceGuo The best way to say thanks here is upvoting and accepting the answer $\endgroup$ – ajotatxe Aug 24 at 3:09
  • $\begingroup$ @Matthew Daly I don't see why does $ (p\wedge q) \to r $ is equivalent to $p\to( q\to r)$ $\endgroup$ – Lawrence Guo yesterday

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.