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Edit based on 2 answers given: The context is that this question was originally part of another question. I believe the lack of distinguishing between the 2 $+$'s is because the instructor wants to further cover up what is actually a trick in the question, but I haven't brought this up to the instructor yet.

Consider a homework question that goes

Let $S$ be the subset of $\mathbb R^n$ s.t. (details details). Define addition and multiplication as follows: For $(a_1,a_2, ..., a_n),(b_1,b_2, ..., b_n) \in S$, define $(a_1,a_2, ..., a_n)+(b_1,b_2, ..., b_n)=(a_1+b_1,a_2+b_2, ..., a_n+b_n)$ and $(a_1,a_2, ..., a_n)(b_1,b_2, ..., b_n)=(a_1 \times b_1,a_2 \times b_2, ..., a_n \times b_n)$, where $+$ and $\times$ are the usual addition and multiplication in $\mathbb R$. Which axioms in the definition of a field are satisfied by $S$? Is $S$ a field and why?

  1. Technically should the $+$ on $\mathbb R^n$ and the $+$ in $\mathbb R$ be distinguished? I mean there's this part 'where $+$', but there are actually 2 $+$'s.

  2. I think it should be something like use a '$+_S$' or '$+_{\mathbb R^n}$' for the vector addition and then regular $+$ for the number addition. Are any of these good?

$(a_1,a_2, ..., a_n)+_S(b_1,b_2, ..., b_n)=(a_1+b_1,a_2+b_2, ..., a_n+b_n)$

or

$(a_1,a_2, ..., a_n)+_{\mathbb R^n}(b_1,b_2, ..., b_n)=(a_1+b_1,a_2+b_2, ..., a_n+b_n)$

BCLC
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    Never ask solely a question based on another question, or you may run into the X-Y problem. Include full the source of the question. And please stop using this site to try to manipulate users into criticizing your instructors. – amWhy Sep 23 '21 at 00:57

3 Answers3

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I think you'll find in most scenarios it is completely acceptable to overload operator notation like this without extra notation, and to look to the context to determine what operation is being used.

The only question is "how likely a misunderstanding is to arise?" If the chances are low, then keep things simple.

There are some situations where two operations on the set deserve to be made distinct this way, but it is not very common.

Even in module theory, for example, an element $m$ of a module and two elements $r,s$ of a ring, it is completely normal to write $mrs$, even though technically the multiplication with the $m$ is a different operation entirely to the multiplication between the $r$ and $s$. And when writing the axiom $m(r+s)=mr+ms$ nobody cares that the two plusses are different operations.

If one faithfully adhered to every technicality notationally, the world would be a lot less readable and a great deal more tedious.

rschwieb
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    'The only question is "how likely a misunderstanding is to arise?" If the chances are low, then keep things simple.' --> Actually this question is a spin-off of something else. The lack of different notation makes me think precisely this is where misunderstanding may arise. What do you think? Thanks – BCLC Sep 21 '21 at 17:52
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    If you had a $+$ for oranges and a $+$ for apples, what would make it hard to recognize which $+$ is meant if both sides of the $+$ have an apple? – rschwieb Sep 21 '21 at 17:54
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    rschwieb ok you know what I think I was unclear. see my edit to OP please. I believe the lack of distinguishing is part of covering up the trick of the question. – BCLC Sep 21 '21 at 17:55
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    What do you think? Honestly, no offense intended, my impression is that this is making a mountain out of a molehill. Of course, that's just perspective, it does not address your problem. In the example you gave, I don't see that the ambiguity is insurmountable. – rschwieb Sep 21 '21 at 18:00
  • re the module, ok fine it's a fair point EXCEPT that students of modules are more advanced. here the students don't even necessarily know group theory. i think it's important to distinguish. and if not, then why even bring up the 'where $+$' ? – BCLC Sep 21 '21 at 18:00
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    @BCLC If introducing a vector space to students for the first time, I might distinguish all the operations as you suggest. But the notation would only last the class and I would fully expect the students to have the hang of the abuse of notation by the end of the week. I might also just say something in passing that "notice these are actually different operations... but you can see how to distinguish them and let's not bother with extra notation." – rschwieb Sep 21 '21 at 18:01
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    rschwieb, well ok i never said it was insurmountable. it's just that it seems like the question is out to trick students who are mere beginners. and i think if you don't have the $+_{\mathbb R^n}$ then you are not being fair in your trick. it's like the Knox's "Ten Commandments" or something. idk – BCLC Sep 21 '21 at 18:02
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    @BCLC By all means, if you are in that teaching position, say something about the difference! Or even use the notation you suggest. But don't use it very long :) – rschwieb Sep 21 '21 at 18:03
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    rschwieb haha wow i sound like a teacher/teaching assistant? I'm just a student. lol. a little overqualified for this 1st half of this class but the 2nd half i indeed didn't study because it wasn't part of the GRE. thanks. now... – BCLC Sep 21 '21 at 18:07
  • 'students for the first time' --> 1 - ok fine. i'll bite. this is the very 1st homework and it was released even before the 1st day of class. What do you think? 2 - Btw, pls look at this part of the lecture notes. It says 'where $a_i+b_i$' instead of just 'where $+$'. What do you think? – BCLC Sep 21 '21 at 18:07
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    Sorry, I have nothing more to say. – rschwieb Sep 21 '21 at 18:19
  • ok thanks anyway. no need to apologise. – BCLC Sep 21 '21 at 18:21
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    @BCLC You seem to misunderstand the applicability of that comic/article. I doubt rschweib intended any form of "thank you", nor was there self-deprecation in that "sorry" that warrants correction. – Mark S. Sep 23 '21 at 01:47
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Yes, this is a standard abuse of notations. The character + is used to describe both the operator $$\mathbb{R}\times \mathbb{R}\to \mathbb{R}$$ sending the pair $(a,b)$ to $a+b$ and the operator $$\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$$ sending the pair $((a_1,...,a_n),(b_1,...,b_n))$ to $(a_1+b_1,...,a_n+b_n)$.

Of course you can also think of it as a map defined on the disjoint union $\mathbb{R}\cup\mathbb{R}^n$, defined in the former way on the subset $\mathbb{R}$ and the latter way on the subset $\mathbb{R}^n$. But then the domain of this function will just keep increasing as you keep doing math.

In practice, so long as there's not chance of confusion, this abuse of notations is standard.

Cronus
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    'In practice, so long as there's not chance of confusion, this abuse of notations is standard.' --> Actually this question is a spin-off of something else. The lack of different notation makes me think precisely this is where misunderstanding may arise. What do you think? Thanks – BCLC Sep 21 '21 at 17:52
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    @BCLC I am not sure that in this case it is the ambiguity of what '+' means that led to the confusion you described in the other question. In any case, it is certainly the case that sometimes some people will consider an abuse of notations confusing, while others would not. I haven't read the question you linked to thoroughly, and it is possible that it was a source of confusion there. – Cronus Sep 21 '21 at 17:58
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    ok you know what I think I was unclear. see my edit to OP please. I believe the lack of Cronus distinguishing is part of covering up the trick of the question. – BCLC Sep 21 '21 at 17:58
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    @BCLC If I understand correctly the trick is asking what axioms two operations fail to satisfy, while in fact their image is not even contained in $S$ as it is supposed to be. But I am not sure writing $+_S$ would make this less confusing. – Cronus Sep 21 '21 at 18:01
  • Cronus perfect understanding! thanks so much! it wouldn't make it less confusing. it would just make the trick more fair/less unfair, like it will give a hint towards that the $+$ in $\mathbb R^n$ is actually really just in $\mathbb R^n$. of course giving a hint will go towards more giving away the trick, but i still think the hint is warranted because without it the trick is really unfair. (with it, the trick is maybe still not fair, but i believe less unfair.) what do you think? – BCLC Sep 21 '21 at 18:18
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    @BCLC I definitely think such abuse of notations should be remarked upon/discussed in the beginning, yes. But perhaps the instructor did not mean to do this intentionally; sometimes one is so used to such abuses of notations, one doesn't even notice them anymore. – Cronus Sep 21 '21 at 18:38
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This is called ad hoc polymorphism, also known as operator overloading. It is a legitimate thing to do from a type theory perspective. See for example How to make ad-hoc polymorphism less ad hoc by Philip Wadler and Stephen Blott.

user76284
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