# Angle brackets for tuples

I've recently noticed that use of angle brackets for writing tuples, e.g. $\langle x, y \rangle$ instead of the usual round brackets in a few books I've been reading — Lawvere's Sets for Mathematics, Mac Lane's Categories for the Working Mathematician, Forster's Logic, induction and sets, for example. I've also seen occasional use of it in Hartshorne's Algebraic Geometry, but there round brackets seem predominant. Is there some subtle distinction between the two notations I've missed, and what might the reasons for not using round brackets be? Is this practice peculiar to a particular tradition in mathematics (say, foundations)?

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I'm also curious, since I've encountered abstract algebra texts using angled brackets, e.g., to denote groups <set, operation>: e.g. $\langle \mathbb Z, + \rangle$. – amWhy Jun 19 '11 at 17:01
In Strooker's Introduction to categories, homological algebra, and sheaf cohomology, he uses $\textopencorner x,y\textcorner$ for tuples, but I haven't seen that anywhere else (edit: the corners doesn't seem to be working, they are on page 9 here). – Zev Chonoles♦ Jun 19 '11 at 18:09

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It may just be a matter of parenthesis being overloaded. – Arturo Magidin Jun 19 '11 at 19:18

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@amWhy: Formally speaking, a structure is a tuple... – Asaf Karagila Nov 10 '11 at 12:10

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– Martin Sleziak Nov 10 '11 at 14:51
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## 4 Answers

I don't think this is quite what you are asking about, but at the level of ordered tuples of elements, I would use parentheses to emphasize that I only care about the tuple as a point object, and angles to emphasize that I care about the tuple as a direction, or vector.

For example, I might have a line parametrized by $t$ with equation$$(x,y,z)=(x_0,y_0,z_0)+t\langle a,b,c\rangle$$

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Is this convention in use by anyone other than you? I have never encountered it, but then I haven't read a lot of mathematical texts. – Rahul Narain Jul 24 '12 at 6:50
@Rahul I don't know. I picked it up somewhere. Stewart uses angle brackets for vectors. But in the vector chapters, points often get angle brackets too. – alex.jordan Jul 25 '12 at 7:10

Some analysts (in a wide sense) write $\langle x,y\rangle$ (angle brackets), for $x$ and $y$ elements of a same set $X$, to denote the ordered pair element of $X\times X$. A more classical (to me) notation is $(x,y)$ (parenthesis), but, if for example $X=\mathbb R$ and $x\leqslant y$, the notation $(x,y)$ may refer to the open interval $\{z\in\mathbb R\mid x<z<y\}$. Hence the bracket notation might have been designed as a way to avoid the confusion. Bourbaki use the notation $]x,y[$ for open intervals and $[x,y]$ for segments. This notation, of frequent use in the mathematical literature written in French (and in others), removes the risk of confusion mentioned above.

I do not know how useful brackets are for objects like $\langle\mathbb Z,+\rangle$, since the objects inside the brackets are of a different nature.

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Kind of @Arturo's point in a comment. – Did Nov 10 '11 at 12:17
What do those analysts (in a wide sense) do in order to avoid confusion of ordered pairs of vectors from a space with a scalar product? Oh, well... There are only finitely many symbols. Further trivium: were taught the Bourbaki-style intervals in elementary school in Switzerland. – t.b. Nov 10 '11 at 12:28
They write scalar products as $x\cdot y$. // Replaced French by written in French. // Thanks for the remarks. – Did Nov 10 '11 at 12:35
– t.b. Nov 10 '11 at 12:49
@t.b. I see... I was not aware of that (and I first thought you were referring to the French speaking part of the country). Thanks. – Did Nov 10 '11 at 13:15

Herbert Enderton --and other logicians-- use angle brackets to indicate a structure, that is, a set with underlying functions, relations and constants. For example, we denote the usual natural numbers with addition, multiplication, an identity 0 for addition and an identity 1 for multiplication (in that order) as $\left\langle\mathbb{N},+^{\mathbb{N}},\cdot^{\mathbb{N}},0^{\mathbb{N}},1^{\mathbb{N}}\right\rangle$.

In general, for a structure $\mathfrak{A}$ with relations $r_1,\dots,r_i$ (each of a given arity), functions $f_1,\dots,f_j$ (each of a given arity) and constants $c_1,\dots,c_k$, an interpretation of this structure with domain $A$ is denoted $\left\langle A,r_1^{\mathfrak{A}},\dots,r_i^{\mathfrak{A}},f_1^{\mathfrak{A}},\dots,f_j^{\mathfrak{A}},c_1^{\mathfrak{A}},\dots,c_k^{\mathfrak{A}}\right\rangle$.

I don't know wheter this notation came from algebra first, but, as commented above, it appears that some authors use angle brackes for algebraic structures in general (e.g. $\langle\mathbb{Z},+\rangle$, $\langle\mathbb{R},+,\cdot\rangle$).

Another use of angle brackets is to indicate an inner product in a vector space, to distinguish the inner product with other well-known operations between the elements of the space, for example, composition of automorphisms of a given set and multiplication of numbers in a ring, which are usually denoted with common syntactic concatenation ($fg$ for the composition of $f$ and $g$, $xy$ for the (ring-)product of $x$ and $y$).

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I'm not an expert, but I'm quite sure the difference between angle and round brackets depends on the individual notation conventions of each author/paper/work.

Sometimes I see angle brackets instead of round ones without any particular meaning, while (in example) in the computability and complexity class I attended only round brackets were used for classic tuples: the angle brackets were used as a shortcut to mean the encoding for a turing machine of that tuple.

Unfortunately I have no clue about the origins of the angle brackets, it would be interesting.

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