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Is one of these phrasings the most clear and official (something you might expect in a published paper)? Is there a better way to phrase this simple statement?

  1. Let $A \in \mathbb{R}$ and $B \in \mathbb{R}$. This seems technically correct, but feels a bit redundant, in particular when the containing space is something long-winded, rather than $\mathbb{R}$

  2. Let $A,B\in \mathbb{R}$. I think that this one is commonly used colloquially, but is a bit sloppy and can be confusing in some contexts.

  3. Let $A$ and $B$ in $\mathbb{R}$. I think the usage of the English word instead of the inclusion symbol seems awkward here.

  4. Let $A$ and $B \in \mathbb{R}$. The English combined with symbols seems a tad unusual to me.

Edit: Most people in the comment section seem to prefer 2). My hesitation here is from reading this mathematical grammar guide by West.

"Lists of size $2$. It is common but ungrammatical to write "Let $x,y$ be vertices in $G$"; we would not write "My friends John, Mary came to dinner." The concatenation is an instance of two formulas separated by a comma. To see what can go wrong, consider the following clause: "Since $a|b$ and $a,b$ are maximal and minimal,". What was meant was: "Since a|b, with $a$ maximal and $b$ minimal,". In general, the comma within a list of two elements should be replaced with "and" when discussing the two elements as individual items. For example, "If $x,y$ are adjacent" should be "If $x$ and $y$ are adjacent" or "If $\{x,y\}$ is a pair of adjacent vertices"."

This example is slightly different from the one I have given in that it does not desire the usage of a symbol (such as inclusion in my example). Those of you who think

  1. is the preferred option, can you rationalize 2) as being acceptable, while agreeing with West? Or do you think West is wrong?
TShiong
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Mark
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    I prefer $2.$ I have never seen $3.$ or $4.$ in any good sources. – Sahiba Arora Jul 08 '20 at 23:21
  • Let $(A,B)\in \Bbb R^2$. – hamam_Abdallah Jul 08 '20 at 23:21
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    The second choice seems perfectly fine to me... in any case, for reasons of grammar, the third choice possibly should be "Let $A$ and $B$ be in $\mathbb R$"... – paul garrett Jul 08 '20 at 23:22
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    I usually read the phrase in 2. That's the one I would use – Andrew Jul 08 '20 at 23:24
  • I can't tell whether West objects to the use of the comma with $\in$, but either way, his comments read as needlessly punctilious. All mathematical notation is governed by convention, and anyone with any mathematical training knows how to parse the string "$A, B\in\mathbb{R}$" without lapsing into grammatical infelicity. – symplectomorphic Jul 08 '20 at 23:44
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    About your edit, a comma in mathematical language is always interpreted as "and". So when it is written "let $A, B \in \mathbb{R},$" it is read as "let $A$ and $B$ belong to $\mathbb{R}$". For the example given by West, it is worthwhile to note that one would always find it written: "Since $a \mid b$ and $a,b$ are maximal and minimal respectively". If the sentence lacks "respectively", it does create doubt and lacks clarity. – Sahiba Arora Jul 08 '20 at 23:49
  • 2 is definitely the most common, but I really dislike it personally for the exact reason that 1 makes more sense, if you know what I mean. I have never ever seen 3 nor 4. – gen-ℤ ready to perish Jul 09 '20 at 00:16
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    I vote for 2. I think the meaning is clear and cannot imagine a non contrived situation where it is ambiguous. The goal is to clearly communicate to other readers not pedantic correctness. – copper.hat Jul 09 '20 at 00:42
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    A fifth option is ${A, B}\subset\Bbb R$ also – gen-ℤ ready to perish Jul 09 '20 at 00:44
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    About the edit: Your observation about West’s example not including a symbol really is the reason for one being acceptable and the other less so. In general, words in the text cannot interrupt formulas. So, “Let $x$ and $y \in \mathbb{R}$” would parse (at first) as “Let $x$ and let $y \in \mathbb{R}$”, which is strange as thus avoided. On the other hand, in “Let $x, y \in \mathbb{R}$”, the complete part “$x, y \in \mathbb{R}$” is read as one formula, so English grammar rules do not apply within it. (If you wish, you might read it as “Let it be true that $x, y \in \mathbb{R}”, [cont] – Eike Schulte Jul 09 '20 at 06:51
  • [cont] though most people rather translate it into “Let $x$ and $y$ be element of $\mathbb{R}$ in their head, I guess.) In West’s example “Let $x,y$ be vertices in $G$” the situation is different because “$x, y$” alone does not parse as a formula (at least for me). The variables are “connected” to the graph $G$ by English (instead of “$\in$”) anyway, so now English grammar applies and dictates the use of “and”. – Eike Schulte Jul 09 '20 at 06:55
  • (2) is the best of your options, but all of them are lousy; see this explanation. – user21820 May 24 '22 at 09:47

2 Answers2

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As the commenters say, option 2 is standard. But the simplest thing to say is just, "Suppose $A$ and $B$ are real numbers." In many cases the formalism may well be unnecessary and out of place.

symplectomorphic
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    +1 for recommending words instead of symbols. In addition, I'd prefer lower case $a$ and $b$. Upper case suggests subsets. – Ethan Bolker Jul 08 '20 at 23:44
  • @EthanBolker: yes, I would use lowercase too for real numbers -- I was just preserving the rest of the sentence as given. – symplectomorphic Jul 08 '20 at 23:45
  • This makes sense. I just chose the reals as a simple example, but the case I am interested is quite complicated notation-wise and would not be easily translated to English. – Mark Jul 09 '20 at 00:14
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    @Mark: Consider, then, defining a set $S$ in symbols, and then saying "Let $a$ and $b$ be elements of $S$." It is a good exercise to write in natural language as much as possible. (I find it often forces me to articulate ideas I wouldn't otherwise have thought of trying to express.) – symplectomorphic Jul 09 '20 at 04:17
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One alternative is to use cartesian product.

For instance, instead of

Let $A\in E, B\in F $ and $ C \in G$,

it is more correct to write Let $ (A,B,C)\in E\times F \times G$.

hamam_Abdallah
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    I think Cartesian products make you think of $(A,B,C)$ as a single object, which may not confuse your point. – Mark Jul 08 '20 at 23:28
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    “The best” — Well, it’s an option. I’d consider it an alternative really. I think the superlative is what’s preventing this from getting more upvotes, like one from me for instance. – gen-ℤ ready to perish Jul 09 '20 at 00:39