I do not understand the definition of antisymmetric relations OK, let A be a set and let R be a binary relation on A. In my class we say that R is antisymmetric if and only if for every a, b in A, if (a, b) in R and (b, a) in R then a = b.  Fair enough, but what does "a = b" mean? Does this mean that A has some sort of a priori equivalence relation that allows me to identify a and b?
I would find this surprising, in what I've read about partially-ordered sets, I've never seen any discussion of an underlying equivalence relation.
Thanks for enlightening me!
Bob Muller

In the language of mathematics, $=$ is the "equality" relation. Two things are equal if and only if they are the same thing, in which case we write it as $a=b$.
How can two things be the same? Well, one can perhaps define something in two ways, and then we argue that the two definitions give rise to the same object. For example $A$ is $1+1$ and $B$ is $\sqrt4$.
Therefore $R$ is antisymmetric if whenever $(a,b)\in R$ and $(b,a)\in R$ then $a=b$, that is $(a,b)=(b,a)=(a,a)=(b,b)$.
