15

I am trying to write a small article, and I just want to know how would be a good way to present the maths I have written so that it looks professional.

I am trying to define a sequence $x_n$ of real numbers. So what I wrote in my article is:

Let $x_n$, $n \in \mathbb{N}$ be a sequence of real numbers.

However, it does not look very professional. How would I write the above sentence into something that looks professional? Note: I need to include $n \in \mathbb{N}$ in my sentence, so I think thats where my trouble is as $x_n$, $n \in \mathbb{N}$ seems a bit messy.

Thanks.

Gary
  • 411
  • 1
  • 4
  • 7
  • 19
    I prefer to present a sequence as $(x_n){n\in \mathbb{N}}$ (the parentheses are used to mean that a sequence is "ordered" in some way, just like finite $n$-tuples do). Other people use also ${x_n}{n\in \mathbb{N}}$ (but I don't like this symbol, because curly brackets always denote sets and sets are not necessarily "ordered"). – Pacciu Mar 03 '12 at 15:29
  • Is it an infinite sequence? Then maybe let $x_n$, where $n$ ranges over $\mathbb{N}$, be real numbers. Using more symbols is not necessarily more professional. – André Nicolas Mar 03 '12 at 15:31
  • Thanks for your help guys! Yes it is an infinite sequence. – Gary Mar 03 '12 at 15:47
  • Just to add to the previous answers, in the Z formal specification language, curly brackets denote a set and angle brackets denote a sequence. copied from this link (3rd Paragraph) – Alma Rahat May 09 '13 at 14:40

2 Answers2

10

I write my sequences as $\langle x_n \mid n \in \mathbb{N} \rangle$. Looks pretty cool I think. So you could say let $\langle x_n \mid n \in \mathbb{N} \rangle$ be a sequence in $\mathbb{R}$.

Paul Slevin
  • 4,631
  • 11
    Notation $\langle \cdot \mid \cdot \rangle$ is used by physicists to denote inner product in the sequences space $\ell^2$; also, most mathematicians use the angular brackets $\langle \cdot ,\cdot \rangle$ to denote the inner product in a Hilbert space (hence also in $\ell^2$). Hence, IMHO, the notation you use can be very ambiguous. – Pacciu Mar 03 '12 at 17:41
  • 3
    I probably use this more since this is the standard notation for sequences in set theory. – Paul Slevin Mar 03 '12 at 18:07
  • 2
    I think it's not too ambiguous in this case. Probably no one would denote an inner product with $n \in \mathbb{N}$ in one of the sides of the angular brackets. – Gold May 09 '13 at 15:05
  • Big angle brackets is how Lamport denotes sequences in the TLA+ ecosystem. Of course he goes through great pains to make it clear a sequence is formally a function $x: \mathbb{N}\rightarrow S$ – Michael Deardeuff Sep 13 '13 at 22:26
  • @Michael Deardeuff: In TLA+, angle brackets denote tuples, which are finite sequences. For example, $\langle a, b \rangle$. In other words, they are "tuple constructor" syntax. They cannot be used to define infinite sequences, so they cannot be used to answer the OP. Instead, one has to define a function $f$ with $\mathrm{DOMAIN}\ f = \mathit{Nat}$. – 0 _ Dec 10 '16 at 09:15
  • What notation would one use to denote a variable to which a sequence is assigned? Capital letter (e.g. A = ⟨a,b⟩)? Lowercase letter? Bold? Italic? Set variables are often capital letters. Some sets have special styling for their capital letters (as seen in this answer for natural and real numbers). Thoughts? (Should I break this out into a new question?) – Jacob Barnard Jun 29 '23 at 16:33
10

The best notation for functions that I have encountered so far is by Leslie Lamport, described in his book on the temporal logic of actions, TLA+ (see Sec.5.2 on p.48, and Sec.6.4 on p.69).

In axiomatic set theory as formalized in TLA+, a function is a set that satisfies certain axioms about relevant syntactic constructs. I have found the syntax exceptionally practical and precise:

  • $f \triangleq [x \in S \mapsto e(x)]$ defines a function that maps elements of the set $S$ to the expression $e(x)$. Example of function constructor syntax: $f \triangleq [x \in \{3, 4\} \mapsto x - 1]$

  • $\mathrm{DOMAIN}\ f$ is the domain of function $f$. If $f$ is defined as above, then $\mathrm{DOMAIN}\ f = \{3, 4\}$

  • $f[x]$ function application, is a known value only if $f$ is a function (an axiom schema tells us that whatever is defined using the function constructor syntax from above is a function). Otherwise, it is some unspecified value (value is synonym for set in the set theory underlying TLA+).

  • $g(x)$ is operator application syntax, not function application. Read Sec.6.4. A function is a set, so we can write $\exists\ f$. An operator $g$ is syntax. Only the value $g(x)$ that results from applying the operator $g$ to the argument $x$ is a set.

    This is the only approach that I have found across the mathematics, computer science, and engineering literature that addresses the question of what "f[x]" means when $x \notin \mathrm{DOMAIN}\ f$. To see what the problem is with traditional notation and education, see this discussion.

  • $f \in [S \rightarrow T]$ says that $f$ is an element of the set of functions with $\mathrm{DOMAIN}$ the set $S$ that take values in the set $T$. Note that "codomain" is not part of the notion of function. Only the range of a particular function makes sense, and the $T$ used to define some specific set of functions.

Using the above notation for the case of functions from the natural numbers (OP), a sequence of real numbers is a function

$f \in [\mathbb{N} \rightarrow \mathbb{R}]$

OLD ANSWER:

The notation $\{x_n\}_{n\in\mathbb{N} }$ is appropriate because a sequence is formally a function $x: \mathbb{N}\rightarrow S$ that maps natural numbers to elements of the set $S$ (codomain). The index $n$ denotes the argument of function $x$. The first notation corresponds to a set of numbers $\{x_n\}$ (the curly brackets do denote a set) indexed by the natural numbers.

Note also that the space of all real-valued sequences can be denoted as $\mathbb{R}^\mathbb{N}$, i.e., the set of all functions from the natural numbers $\mathbb{N}$ to the real numbers $\mathbb{R}$. See also this discussion: What does it mean when a set is the exponent?

Community
  • 1
0 _
  • 367
  • Doesn't this cause problems for the order of the elements in the sequence? idea from this other question – Mr Tsjolder from codidact Jan 15 '17 at 16:12
  • The order of the elements in the sequence is the order defined by the linear order between natural numbers. $f[0]$ is the first element, $f[1]$ the second, etc. If you prefer to start counting from 1, then just use $\mathbb{N} \setminus {0}$ as the function's domain: $f \in [\mathbb{N} \setminus {0} \rightarrow \mathbb{R}]$. For example, in TLA$^{+}$, a tuple is any function indexed over the set $1..n$, for example $\langle 6, 15 \rangle$ is the function $[x \in {1, 2} \mapsto \mathrm{IF}; x = 1; \mathrm{THEN}; 6; \mathrm{ELSE}; 15$. – 0 _ Jan 16 '17 at 05:47
  • For the function-notation, there is no problem. I rather meant the set-notation. – Mr Tsjolder from codidact Jan 16 '17 at 11:06
  • 1
    Well, the notation that looks similar to a set constructor has the $n \in \mathbb{N}$ outside the braces, so it is distinct from ${x_n : n \in \mathbb{N}}$. It could be recognized as different from a set constructor, but the difference is small. However, in any case it is informal, so it's meaning is unclear. In contrast, the TLA$^+$ notation is formally defined with axioms, so there isn't any ambiguity there. This is the problem with informal mathematics. – 0 _ Jan 16 '17 at 17:23