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This question is about using correct notation while writing a research paper. Say I have a directed graph $G$, partitioned into $T$ layers. Denote the set of nodes in layer $t$ by $V_t$. Suppose, I wanted to write some constraint, one for each node in $G$, then which among the following notations below are correct.

a. Constraint involving $v$, $\forall t \in [T]$, $\forall v \in V_t$.

b. Constraint involving $v$, $\forall t \in [T]$, $v \in V_t$.

c. Constraint involving $v$, $\forall v \in V_t$, $\forall t \in [T]$.

d. Constraint involving $v$, $\forall v \in V_t$, $t \in [T]$.

Basically, think of the "Constraint" above as the flow balance constraint that we would like to enforce at every node in our graph.

PS: I have seen papers with notation a, b, c, d and am somewhat confused as to which one is correct \ standard notation. If you feel this may not be the correct forum to ask this question, I would be happy if you can redirect me to the appropriate stack exchange.

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This is a matter of personal preference, to some extent. Personally, I like a and b more than c and d, because I feel that $t$ is "out of scope" in c and d; in other words, you're using $t$ before you introduce it. I would tend to think of c and d as slightly incorrect notation, though not egregiously so.

In terms of a vs. b, that is definitely a matter of preference—neither is right or wrong. I have used both.

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I think the first three are OK, more so the first two. I agree with what Larry said about defining all variables first before usage and in this case it means defining $t\in[T]$ first.

However, the last one not only has the minor problem above, but it also carries some ambiguity, as $\forall v\in V_t,t\in[T]$ can mean either:

  • for all $v$ in $V_t$, and for all $t$ in $[T]$, or

  • for all $v$ in $V_t$, where $t$ is in $[T]$.

The latter misses the 'for all' part, although I believe most people would read it in the first way.

Note that this does not apply to the second constraint as it is evident that $v\in V_t$ is incorporated within $\forall$.

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I would personally go for readability over mathematical stenography:

Constraint involving any $v:v\in V_t$, where $t\in [T]$.

For some people, $\forall$ has subtly different implications than simply drawing values from a set, as it explicitly says that the statement must be true for every single element in the set. Most of the time however people will treat the two notations as interchangeable.

A nice example is:

$\log(x),x\in \mathbb{R}$, vs $\log(x),x\in \mathbb{R}^+$

Depending on who you talk to, the first statement can be correct or not. The set over which the function is defined is clearly $\mathbb{R}^+$, but the co-domain is all real numbers, so if $x\in\mathbb{R}^+$ then it also belongs to its superset $\mathbb{R}$ by definition.

This is the kind of situation where you know what you mean, they know what you mean, and people sometimes just get annoying about terminology and correctness. I have found that using plain English where appropriate can make your math far more concise and explicit, so feel free to do so if that makes sense to you.

Remember, mathematical stenography is just that: shorthands so that we don't need to write words. Early mathematical texts didn't contain any symbols and the logic was conveyed just fine.

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