I do not understand the definition of antisymmetric relations

Question

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

Answer 1

Not only an equivalence relation on a non-empty set $A$ defines a partition of $A$ in equivalence classes, the reverse is also true: every partition of $A$ corresponds to an equivalence relation. Hence the partition which arises from the singletons of $A$ corresponds to an equivalence relation $R_{\text{singleton}}$. The usual name for “$R_{\text{singleton}}$” is “$=$”, i.e., $aR_{\text{singleton}}b\iff a=b$. Enlightened enough?

Answer 2

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 $(a,b)=(b,a)=(a,a)=(b,b)$.

Answer 3

Perhaps it will make more sense to you if we replace $a = b$ with "it is not the case that $a$ and $b$ are distinct." Afterall, we define two elements as being distinct by asserting $a \neq b$. The negation of $a\neq b$ is the negation of "a and b are distinct" which is thus $\lnot (a \neq b), \text{ i.e., } a = b$.

In short, you can think of asymmetry as satisfying, for all $a, b$ in your set, the implication: $$[(a, b)\land (b, a)] \implies \underbrace{a\text{ and } b\;\text{are not distinct}}_{\large a\,=\,b}.$$

Answer 4

There's no trick here, the $=$ is literal identity. Note that the antecedent of the antisymmetry condition does not say that $a,b$ are distinct. For any element reflexively related by $R$, e.g. $aRa$, that element is trivially related symmetrically to itself. Antisymmetry says that the reflexively related elements are the only symmetrically related elements.

If you think of the partial ordering as being some kind of ranking system, antisymmetry says the only person you can be "tied with" is yourself.