Distinguishing the $+$ in $\mathbb R^n$ from the $+$ in $\mathbb R$ (where NOT distinguishing MAY INDEED cause confusion)

Question

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.

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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

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

Answer 1

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.

Answer 2

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.

Answer 3

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.