I'm reading up on "Set Theory and Logic" by Stoll and came upon notation for relations that I haven't seen before. I've seen $x\sim{y},$ and $xRy$ before but Stoll uses this one.
$$(x,y)\in{\rho}$$
Now I admit that the prior two are specifically binary relations and I haven't seen ternary or n-ary relations like that. Is the advantage of Stoll's notation apparent through n-ary relations? An example of the ternary relational notation would be
$$(3,5,8)\in{+}$$ where $+$ is addition. Or is this notation not really used? As I mentioned I don't recall seeing this written before but I was curious about its popularity
[Math] Notation for a relation
elementary-set-theorynotationrelations
Best Answer
Formally a relation from a set $X$ to a set $Y$ is a subset of $X\times Y$; if I call this subset $\rho$, the familiar notation $x\mathbin{\rho}y$ is an abbreviation for the more formal $\langle x,y\rangle\in\rho$. Since the most familiar binary relations are typically written with this infix notation (e.g., $x\le y$, $A\supseteq B$, etc.), the more informal $x\mathbin{\rho}y$ is often felt to be more intuitive, but it does obscure the set-theoretic nature of relations a bit.
In short, Stoll’s notation is entirely standard, and you’re likely to encounter it any time you have to deal with relations in the abstract rather than with specific relations.
Actually, $+$ on, say, the integers is a function from $\Bbb Z\times\Bbb Z$ to $\Bbb Z$. Functions are just a special kind of relation, so it’s a relation from $\Bbb Z\times\Bbb Z$ to $\Bbb Z$ and therefore a subset of $(\Bbb Z\times\Bbb Z)\times\Bbb Z$; formally one would write $\big\langle\langle 3,5\rangle,8\big\rangle\in+$, though there is a natural correspondence between $(\Bbb Z\times\Bbb Z)\times\Bbb Z$ and the set of ordered triples of integers.