What is the difference between the notations $j \in \{1, 2, ..., p\}$ and $j=1,2,...,p$? 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?
 A: 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.
A: 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.
