Punctuation in mathematical writing

[jgens]

Recently I went through a bunch of my old solution sets and realized that I've been quite sloppy with my use of punctuation; in particular, examining the quality of my writing in solution sets over a period time, it appears that the lack of proper punctuation is getting more and more pronounced. So, in an effort to fix this trend, I have been trying to use proper punctuation in all of my writing. However, today I came across a sentence that I am uncertain how to punctuate properly, and I was hoping that some of the member of PF could help me out on this one. Here is the sentence:

For each $i \in \{1,\dots,n\}$, write $\alpha_i = b_{i1}\beta_1+\cdots+b_{in}\beta_n$, where $b_{i1},\dots,b_{in} \in \mathbb{Z}$.

My question is whether or not the commas used above are correct and/or necessary. Off the top of my head, I can't think of any rules that necessitate the use of the commas, but when I write the sentence above, I am naturally inclined to use the commas.

Edit: I suppose that since $b_{i1},\dots,b_{in} \in \mathbb{Z}$ is technically a noun, I should write something like "... with $b_{i1},\dots,b_{in} \in \mathbb{Z}$" instead.

 

[DeadOriginal]

I think it looks fine but then again I just started learning how to write proofs not too long ago.

 

[Mark44]

Recently I went through a bunch of my old solution sets and realized that I've been quite sloppy with my use of punctuation; in particular, examining the quality of my writing in solution sets over a period time, it appears that the lack of proper punctuation is getting more and more pronounced. So, in an effort to fix this trend, I have been trying to use proper punctuation in all of my writing. However, today I came across a sentence that I am uncertain how to punctuate properly, and I was hoping that some of the member of PF could help me out on this one. Here is the sentence:

For each $i \in \{1,\dots,n\}$, write $\alpha_i = b_{i1}\beta_1+\cdots+b_{in}\beta_n$, where $b_{i1},\dots,b_{in} \in \mathbb{Z}$.

My question is whether or not the commas used above are correct and/or necessary. Off the top of my head, I can't think of any rules that necessitate the use of the commas, but when I write the sentence above, I am naturally inclined to use the commas.

IMO, the commas are helpful.

Edit: I suppose that since $b_{i1},\dots,b_{in} \in \mathbb{Z}$ is technically a noun, I should write something like "... with $b_{i1},\dots,b_{in} \in \mathbb{Z}$" instead.

No, "$b_{i1},\dots,b_{in} \in \mathbb{Z}$ " is a mathematical statement, which makes it more akin to a declarative sentence than to a noun. The b_is make up the subject, and "$\in Z$" is the predicate. You can use either "where" or "with," and I doubt that anyone would notice, although I lean a bit toward "where."

 

[Fredrik]

It's not so much about grammar as it is about where the reader should make short pauses. In this case, it's natural to make a short pause at both places where you put commas, so I think it would look weird to omit them. "Where" sounds better than "with" to me. "With" sounds better in this rewrite:

For each $i\in\{1,\dots,n\}$, let $\alpha_i$ be a linear combination $b_{i1}\beta_1+\cdots+b_{in}\beta_n$ with $b_{i1},\dots,b_{in} \in \mathbb{Z}$.

I assume that's what you meant. Here I prefer to not use a comma before the "with".