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)?
Angle brackets for tuples
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0I've come across angle brackets around tuples to describe a Markov Reward Process, see slide 10 in these [lecture notes](http://www0.cs.ucl.ac.uk/staff/d.silver/web/Teaching_files/MDP.pdf) – 2017-08-09
5 Answers
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
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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0@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. – 2011-11-10
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.
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 $y$).
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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0@Rahul I don't know. I picked it up somewhe$r$e. Stewart uses angle brackets for vectors. But in the vector chapters, points often get angle brackets too. – 2012-07-25
I have just been having the same conundrum. Been reading Hrbacek and Jech's book introduction to set theory and they don't specify exactly what the angled bracket notation means, however from context it became clear once I read through more of the book that they use it to represent a sequence of elements, not necessarily a set.