# How to prove two definitions of antisymmetric are equivalent

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?

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 $$(p\wedge \neg r)\to \neg q$$.
• @Matthew Daly I don't see why does $(p\wedge q) \to r$ is equivalent to $p\to( q\to r)$ – Lawrence Guo yesterday