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first of all, I am very grateful for any help! Please don't take offense if the question is trivial. However, I am primarily a business student and therefore have little prior experience in neat notation of mathematical formulas. I am currently writing my thesis and want to ensure that the notation is consistent.

Example 1:

Suppose we have $p$ variables in the model and one of the variable is given by $X_j$. What is the correct notation for index $j$?

a) $j \in \{1, 2, ..., p\}$

b) $j=1,2,...,p$

Example 2:

Suppose we have a time index $t$ where $t$ can take the values $t_1, t_2, ..., t_P$.

What is the correct notation for index $t$?

a) $t \in \{t_1, t_2, ..., t_P\}$

b) $t=t_1,t_2,...,t_P$

I currently prefer alternative a). However, the literature differentiates between the two versions depending on the paper. Can someone tell me where exactly the difference in notation is?

RobPratt
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Desperate_undergraduate
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    Both notations are correct in each case. Pick the one you like (or the one most common in papers you refer to). Just be consistent. If you are really worried you could define a LaTeX macro for the construction and postpone deciding which format you want until you're all done. – Ethan Bolker Aug 15 '22 at 19:17

2 Answers2

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The notation $j \in \{1, 2, \dots, p\}$ is very formal. The notation $j = 1, 2, \dots, p$ is less formal, and some people may object that it's not even entirely correct: what kind of an object is "$1,2,\dots,p$" and how can $j$ be equal to it? But either one will be understood.

There are alternatives, too. Often (especially in combinatorics) the notation $[p]$ is used for the set $\{1,2, \dots, p\}$ and so we may simply write $j \in [p]$. Also, if it is already clear that $j$ is an integer, we may write $1 \le j \le p$. (In the second example, we might write "$t = t_i$ for some $1 \le i \le P$".)

Be consistent about your notation, to avoid confusing your reader, and don't be afraid to say what you mean in words rather than symbols.

Misha Lavrov
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    An additional thing to consider: often $i,j,k,l,m,n$ are interpreted as integers by default, while $a,b,c,r,t,x,y$ are usually real numbers, and $w,z$ are often complex numbers. However, those are very conventional and vary by field, so it's best to be explicit! – kc9jud Aug 15 '22 at 19:35
  • Thank you very much for your help! – Desperate_undergraduate Aug 15 '22 at 21:46
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The comments and answers so far aren't getting across one way in which they are different.

Taken in isolation, something like "$j \in \{1,\dots,p\}$" or "for $j \in \{1,\dots,p\}$ is ambiguous as to whether or not you mean "for every" or "for some".

But if I say "for $j = 1,\dots,p$ this is essentially always "for every"; it is shorthand $p$ statements, one for each possible value of $j$.

e.g. (I'm trying to come up with bad/ambiguous writing here:)

  • "A monomial is a function $f(x) = x^j$ for $j \in \mathbb{N}$."
  • "A monomial is a function $f(x) = x^j$ for $j \in \{1,2,3,\dots\}$"
  • "A monomial is a function $f(x) = x^j$ for $j = 1,2,3,\dots$"

None of these is good writing. In my opinion this is progressively worse though: The first one is maybe just about OK but most people would agree that the third one is actively wrong: The "$j = ...$ implicitly says "for every $j$" which makes it nonsensical that $f$ doesn't depend on $j$.

One could probably think of other subtly ambiguous situations. Agree with Misha, you just need to try to practice good writing all the time; there aren't really hard and fast rules.

SBK
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