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I'm a maths student in Germany and I am currently writing my thesis in English. Now I want to say the following:

Let A and B denote the first and the last point respectively in the sequence S such that A and B belong to C

So A is the first point and B is the last point ...

I am really not sure about where I should put the word respectively. Also I'm unsure about where I should put commas.

Thanks in advance.

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2 Answers 2

I think you've placed the word correctly. 'Respectively' is used after a pair of lists, to indicate that the items in the lists line up with each other - the first item of one list goes with the first item of the second, etc. It should generally come just after the second list of things, thus:

The values of A, B, and C are 5, 8, and 2, respectively.

I would set it off with commas before and after, as suggested here, but there seems to be some debate on this point. This style manual says that 'respectively' should be set off with a comma when it comes at the end of a sentence, but doesn't mention what to do when it's in the middle.

So, I would write the sentence:

Let A and B denote the first and last points, respectively, of sequence S...

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So you would write ... the first and last points ... instead of ... the first and the last point ... ? Should we use plural points here? Thanks for your answer. –  Stefan Hamcke Aug 25 at 15:01
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I think either 'the first and last points' or 'the first point and the last point' are correct, but the former sounds more natural to me. 'the first and the last point' sounds odd, but I'm not sure if it's actually incorrect. –  MrTheWalrus Aug 25 at 15:03
    
@MrTheWalrus I would imagine first and last point to be incorrect, since it would be using a singular noun, namely point to refer to two such instances. –  Tyzoid Aug 25 at 17:31
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@MrTheWalrus "The first and last point" would refer to the single point that is both first and last. –  David Richerby Aug 25 at 17:39
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@StefanHamcke Yes, "the first and the last point" works perfectly. –  David Richerby Aug 25 at 23:11

The natural phrasings are:

  • "let A and B be X and Y, respectively";
  • "let A and B, respectively, be X and Y";
  • "let A and B be, respectively, X and Y"; and
  • "respectively, let A and B be X and Y".

Your phrasing is OK but seems less natural, since you've put "respectively" into the middle of the long phrase that defines what A and B are (equivalent to "X and Y" in my abbreviated examples).

Also, I suspect you actually mean:

  • Let A and B, respectively, be the first and last points in the sequence S that are in C.

Your sentence means "A is the first point of the sequence and B is the last. Also, I'm telling you that A and B are in C." My version means, "A is the first point of the sequence that is in C; B is the last point of the sequence that is in C."

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No, I actually want to say "A is the first point of S that is in C" and "B is the last point of S that is in C". First and last refer to "point in S which is also in C". –  Stefan Hamcke Aug 25 at 22:34
    
@StefanHamcke Adding "such that" changes the meaning: in this case, it essentially means "The effect of all this is that". The sentence you have in the question means "A is the first point of S; B is the last point of S. The effect of all of this is that A and B are in C." You only use "such that" when you want to say something like "The first x such that f(x) has some property", rather than "the first x such that x has the property", if that makes sense (the second one should be "the first x that has the property"). –  David Richerby Aug 25 at 23:20
    
I think in maths we use "such that" in order to restrict the properties of the defined object, see mathematics-in-europe.eu/en/home/65-mathhelp/… or mathworld.wolfram.com/SuchThat.html. If the first and last elements of the S necessarily belonged to C, then I would maybe use: "... in S, so that A and B belong to C." –  Stefan Hamcke Aug 26 at 14:45
    
@StefanHamcke Yes, we use "such that" to restrict properties. But it's always of the form "x such that f(x) has some property", not "x such that x has some property". If you write "x such that x", it looks like you meant to write "x, such that x", which means "x so that x". (Native-speaker and mathematician, here.) –  David Richerby Aug 26 at 14:58
    
Oh, you're a mathematician, too :-) I'm still not sure about the wording, maybe I'll change it to "Let i and j denote the first and the last index such that v_i and v_j belong to C." Anyway, let's just assume I need some sort of such that in my sentence. Is the placing of respectively still okay? Or does your advice about the placing still apply then? –  Stefan Hamcke Aug 26 at 15:25

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