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Standard definition of writing interval states that it should be written (a,b) where a<b

Due to this being arbitrary and just a convention that we all use, would it be considered a mistake to write it with non standard notation (b,a) where b>a?

I am interested in hearing opinions of those who teach/grade students.In my opinion points should not be deducted for writing interval this way if the student is able to explain the reasoning behind it.

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    $\begingroup$ The way you describe it, there's no reasoning behind it and there couldn't be, as it's just a convention. Does the student explicitly introduce the conflicting non-standard notation? But first of all, why? Just to make the grader angry? $\endgroup$ May 1, 2022 at 11:03
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    $\begingroup$ I'm not sure why you say "both notations can be used", since you didn't say that there was a reason this non-standard notation was usable. $\endgroup$
    – Sue VanHattum
    May 1, 2022 at 16:11
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    $\begingroup$ Unless I were told explicitly about the alternate notation (and I'd wonder why use it, and if it were a paper or book, that would be a huge red flag for me continuing to read/browse the item), I would assume something like $(3,2)$ is equal to the empty set. Indeed, sometimes it's helpful to have this convention to avoid special cases in expressions and theorem statements, although authors should be explicit about this convention also. $\endgroup$ May 1, 2022 at 17:02
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    $\begingroup$ "The student knows both notations can be used..." The student is wrong; they cannot both be used. $\endgroup$ May 2, 2022 at 16:14
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    $\begingroup$ Such a choice is sort-of like driving "on the wrong side of the road": it's not abstractly "wrong", but depending whether you're in the UK v.s. US it has vast practical ramifications. $\endgroup$ May 4, 2022 at 1:38

5 Answers 5

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The goal of notation is to clearly and precisely communicate an idea. The notation $$ (a,b) = \{ x : b < x < a \} $$ fails to clearly communicate an idea, as this notation is unusual, and conflicts with the the ordinary notation for an open interval. As such, I would avoid this notation in the setting of precalculus, calculus, and real analysis, and would deduct (a small number of) points from a student who used this ambiguous notation.

It might, however, be worth noting that the notation $[\alpha, \beta]$ is sometimes used in complex analysis to denote the segment $$ [\alpha, \beta] = \{ (1-t)\alpha + t\beta : 0 \le t \le 1 \} \subseteq \mathbb{C}, $$ i.e. a parameterized, straight-line path from $\alpha \in \mathbb{C}$ to $\beta\in \mathbb{C}$. In this context, it is possible for $\alpha$ and $\beta$ to both be real numbers with $\alpha > \beta$. Even in this case, though, the order in which the two numbers are written matters: the segment is oriented, in that the "initial point" is $\alpha$, and the "terminal point" is $\beta$.

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In theory you could choose to define a notation such as $$(a, b) = \{x ; b < x < a\},$$ but why would you do it? It has exactly the same power as the usual notation, and now the reader either has to remind themself what you mean all the time or make mistakes due to going against a convention.

If there is a wider context where this notation is more in harmony with other notation of the subfield, then there is some justification for it. There might be some other niche situations.

This is especially bad since $(a, b)$ might be an non-empty interval or an empty interval depending on the convention, and both cases might come up in a calculation or a proof.

At the minimum I would advise using some alternate notation for open intervals, such as $]a, b[$, for the reversed interval notation. But again, what is the benefit of breaking convention?

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  • $\begingroup$ I understand that there is no benefit but as i wrote in the comment above. Quote: There was no any intention, the student knows both are correct and writes (a,b) where a>b due to added - in front of b.Originally it was written as (a,b) where a<b, but later on the minus itself changed the notation.The student knows both notations can be used so doesn't swap places due to hypothetically speaking not being allowed to correct it.Would you consider this a mistake as an educator yourself? $\endgroup$ May 1, 2022 at 12:24
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    $\begingroup$ Please edit the extra information to be a part of the question. I am not sure what a minus sign does here; maybe you can clarify that as a part of your edit. $\endgroup$
    – Tommi
    May 1, 2022 at 17:28
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    $\begingroup$ Again, it's surely not that the student "knows that both notations can be used". To use that notation is to perversely sabotage one's communication... $\endgroup$ May 2, 2022 at 3:11
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$(a,b)$ has the advantage of matching how we would order numbers on the number line. It would be possible to have $(a,b)$ for the open set between $a$ and $b$ , but why not order them?

Conventions matter a lot, otherwise how will we communicate effectively? When awarding points/scores/marks deciding if a student understands the standard conventions of the subject is an important component. Indeed, when they write something and expect me to understand it I have to assume they are following the convention. Do you want the "Humpty Dumpty" defense:

“When I use a word,” Humpty Dumpty said in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.” “The question is,” said Alice, “whether you can make words mean so many different things.”

I would deduct some marks for failing to follow the standard conventions under most circumstances. Of course, we have a responsibility to explain the difference between a notational convention and when we overload notation. E.g. $x^{1/2}$ doesn't make sense in terms of repeated multiplication ($x^n$ is $x$ multiplied $n$ times). Here we extend exponential notation to rational exponents, in a way which tries to preserve the theorems, e.g. $x^nx^m=x^{n+m}$. This is quite different from a pure convention. It is the "principle of permanence of forms". There is a real blurring of the difference between pure convention and how analogy is used to choose notation or interpret notation. I wished we talked about this more!

There are some interesting and important cultural differences in the way people write intervals. I quite like $ ]0,1] $ for $ 0<x \mbox{ and } x\leq 1$ but that isn't what I normally write!

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Although I agree that this is terrible notation (given the standard notational expectations), there is a subtle point to address here.

In my opinion points should not be deducted for writing interval this way if the student is able to explain the reasoning behind it.

This is one of those rare (in my opinion) situations where one may be justified in giving "clawback" points to a student who pushes back against being marked incorrect if they can clearly explain what their notation means. The following must be observed here:

  • One is teaching in a cultural context where "partial credit" is a typical expectation.
  • The problem was marked wrong for a nontrivial number of points. This depends on context - if it's a very minor part of a huge question one might overlook it completely.
  • The student asks about why it was marked wrong.
  • The instructor asks the student to explain the notation.
  • The student clearly explains the notation as in the original question.
  • The instructor then explains why this notation either should mean the empty set or would mean an oriented interval (see below, but then again oriented intervals aren't equal).
  • The student shows understanding.

Oriented intervals? Sure, considered as a "path integral"/"vector line integral", $\int_{(2,3)}1\cdot dx=\int_2^3 1\, dx=1$ but $\int_{-(2,3)}1\cdot dx=\int_3^2 1\, dx=-1$, where I consider here $-(2,3)$ as a shorthand for "the interval as a curve, but parametrized to go in the opposite direction". (I think these are both standard notation in what used to be called "advanced calculus", see e.g. here.) The FTC/Stokes' Theorem is even true here, if one has an oriented boundary (well, and if one allows for a boundary of a non-closed interval, and ...). And the student might enjoy seeing a place where this notation could be used, while saying that for now we do not do so as it introduces unacceptable complexity.

Having said all that, I think there is an even more basic reason for not accepting this notation; there is already (unacceptable?) ambiguity in notation between the point $(2,3)\in \mathbb{R}^2$ and the interval $(2,3)$, and it is just increased by allowing this notation.

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  • $\begingroup$ The interval $-(2,3)$ is, in most contexts, the same as $(-3,-2)$ (more generally, if $A\subseteq \mathbb{R}$, then $$-A = \{ -x : x\in A\},$$ which is not the same as just changing the orientation of an interval). Integrating over sets also typically implies that it is the Lebesgue integral which is being considered, rather than the Riemann integral, and there is no real notion of "orientation" for the Lebesgue integral. I will agree, however, that if $\alpha,\beta\in\mathbb{C}$, then $$\int_{[\alpha,\beta]} f(z)\,\mathrm{d}z = -\int_{[\beta,\alpha]} f(z) \,\mathrm{d}z.$$ $\endgroup$
    – Xander Henderson
    May 5, 2022 at 16:50
  • $\begingroup$ But in complex analysis, it is well understood that $\int_{\gamma}$ denotes the integral over some parameterized curve. It has been a while since I've thought about complex analysis, but you get something like $$\int_{\gamma}\,\mathrm{d}z = \int_{a}^{b} \gamma'\,\mathrm{d}t, $$ where $\gamma : [a,b] \to \mathbb{C}$ is a continuously differentiable curve. $\endgroup$
    – Xander Henderson
    May 5, 2022 at 16:52
  • $\begingroup$ It's presumably been even longer since I thought about line integrals, but that is what I was thinking of. I'll try to clarify my answer. $\endgroup$
    – kcrisman
    May 5, 2022 at 23:51
  • $\begingroup$ Hopefully clarified previous sloppiness on "line" versus "path" integrals. $\endgroup$
    – kcrisman
    May 6, 2022 at 0:07
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    $\begingroup$ We have different standard words for the different concepts of "path" vs. "interval", and I'm not sure it's helpful to say that sometimes they can be mashed together in any way. $\endgroup$ May 6, 2022 at 2:20
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I just looked this up in a book of mine (Thomas Finney, 1980). The convention is all greater than the first (a) and smaller than the second (b). So, if your first number is greater than the second one, the result is the empty set, not just some backwards-directed interval. This is not just arbitrary. It allows doing various manipulations.

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  • $\begingroup$ This is an argument from authority with no rationale given for why this convention matters, nor any acknowledgement that other conventions exist and are useful. Yes, one can define an interval $(a,b)$ with $a > b$ to be the empty set (and this is a useful convention in many contexts), but it is not the only convention, and the notation $(a,b)$ with $a>b$ is ambiguous without further explanation from the student. $\endgroup$
    – Xander Henderson
    May 5, 2022 at 16:46

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