Notation for a relation

Question

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

Answer 1

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.

Answer 2

Relations are represented by sets of ordered pairs; so the advantage of this notation agrees with that representation. In practice it is not all that common - certainly outside of set theory you almost always see infix notation for binary relations. For ternary and higher order relations this gets less practical (I can't think of a single example of "infix-like" notation in a ternary relation right now) and you're more likely to see $R(x,y,z)$ or $(x,y,z) \in R$.