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
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asked Sep 1 '13 at 5:22
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1$\begingroup$ I'd say the advantage of that notation is that you are constantly reminded of what a relation IS (viz. a subset of S x S, for some set S). The elements are order pairs. $\endgroup$ – The Chaz 2.0 Sep 1 '13 at 5:26
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$\begingroup$ I agree with you. I thought it was easier to see that as well but I'm surprised I've never seen it $\endgroup$ – Eleven-Eleven Sep 1 '13 at 5:29
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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.
answered Sep 1 '13 at 5:31
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$\begingroup$ I've only had the basic intro to set theory that every undergraduate gets in Discrete and Abstract/Analysis but this is my first attempt at Set Theory as a field of study. Do you recommend Stoll's book or is there a better one out there? $\endgroup$ – Eleven-Eleven Sep 1 '13 at 5:36
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$\begingroup$ @Christopher: You could do a lot worse for an introduction, though as I recall elementary set theory is just part of Bob’s book. If you already have the book available, I’d use it as a starter and then move up to this, which is excellent and will take you much further into set theory proper. $\endgroup$ – Brian M. Scott Sep 1 '13 at 5:43
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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 $(x,y,z) \in R$.
answered Sep 1 '13 at 5:29
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$\begingroup$ I agree with you as well. I tried to think of the infix notation for at least ternary and I drew a blank. I like the notation but suppose it's in it's home in set theory. $\endgroup$ – Eleven-Eleven Sep 1 '13 at 5:31
