Dear Uncle Colin,

Can you give me an example of an antisymmetric relation?

- Equivalence Sodding Classes, Heaping Endless Rage

Hi, ESCHER, and thanks for your message!

I struggled with the notation for relations for a long while: I didn’t realise that the R in $a\text{ R }b$ stood for something like = or $\subset$.

So a (binary) relation is a way of describing two things and how they relate to each other (hence the name). The symbol in the middle, R, can (usually) be replaced by a clause (e.g. “is equal to” or “is a subset of”) and you’re only allowed to write $a\text{ R }b$ if the resulting sentence is true.

Two is not equal to three, so you may not write 2 = 3.

The integers are a subset of the complex numbers, so you may write $\Z \subset \C$.

Equivalence classes call for symmetric relations – if $a\text{ R }b$ then $b\text{ R }a$ – that is, if $a$ and $b$ are related in one way, $b$ and $a$ are related the same way. The relation = is symmetric (if $a$ is the same as $b$ then $b$ is the same as $a$), but $\subset$ isn’t ($A \subset B$ does not imply $B \subset A$.)

An antisymmetric relation is one where:

  • if $a$ and $b$ are different
  • and you can write $a\text{ R }b$
  • then you cannot write $b \text{ R }a$.

That is, if $a$ is related to a different $b$ in some way, then $b$ is definitely not related to $a$ the same way.

Some examples of an antisymmetric relation:

  • $\ge$ on the real numbers
  • $\subset$ on any set of sets
  • Parenthood (if $a$ is a parent of $b$, then $b$ is not a parent of $a$)
  • $a$ is divisible by $b$ on the integers
  • the empty relation

There are – of course – infinitely many others, but these are ones that I feel give a taste of what the term means.

I hope that helps!

- Uncle Colin