May be my question is simple. But this question is confused me. In fact, I want to present a lecture and I don't want to have grammar mistakes.

My question: which one is correct?

  • A: Let $\mathbb{F}_q$ denote the finite field with $q$ elements. Consider the two elements $\alpha$ and $\beta$ in $\mathbb{F}_q$.

  • B: Let $\mathbb{F}_q$ denote the finite field with $q$ elements. Consider two elements $\alpha$ and $\beta$ in $\mathbb{F}_q$.

In other language, I really like to understand how to use the word the in math writing.


Edited based on the @user2357112 comment.

  • 3
    $\begingroup$ "Consider two elements" means to consider any two elements. But "consider the elements $\alpha$ and $\beta$" refers to special elements. So in your example it is B. $\endgroup$ – Dietrich Burde Jun 3 at 11:51
  • $\begingroup$ Worth remarking: neither phrase implies that $\alpha \neq \beta$. If you intend that, it should be stated explicitly, as in "consider two distinct elements $\alpha, \beta\in \mathbb F_q$". $\endgroup$ – lulu Jun 3 at 11:53
  • 2
    $\begingroup$ As a native English speaker I would have no issue with either in any practical sense (and particularly in the context of academic presentation). If you want a more definitive answer I'd suggest trying the English Language and Usage SE. $\endgroup$ – StephenG Jun 3 at 11:54
  • $\begingroup$ Incidentally, the question "Which of the next sentences is true?" is the wrong question, because neither version A nor version B is a statement with a truth value. You could ask which one is correct, or which one means what you wanted it to mean, but not which one is true. $\endgroup$ – user2357112 Jun 3 at 20:09
  • $\begingroup$ @user2357112 Your reminder is correct. Based on your commnet, the question is edited. Thanks. $\endgroup$ – user3568 Jun 4 at 10:08

A typical statement concerning finite fields would be statement $B$:

Consider two elements $\alpha$ and $\beta$ in $\Bbb F_q$.

Here the two elements are arbitrary. With an article "the two elements" they would be specific elements, such as $0$ and $1$ for example. Usually such a specification would then be given. "Let $\alpha$ and $\beta$ be the elements in $\Bbb F_q$ given by the sum of all squares respectively of all cubes."

  • $\begingroup$ Excuse me that I asked too questions. My last question is that "pairwise distinct elements" is true or "distinct pairwise elements"? Thanks $\endgroup$ – user3568 Jun 3 at 12:10
  • $\begingroup$ Pairwise distinct elements. $\endgroup$ – Dietrich Burde Jun 3 at 12:11
  • $\begingroup$ @user3568 Also, "too" here is incorrect, it should be "two". "Too" means "also", while "two" is the number "2". $\endgroup$ – Ovi Jun 3 at 12:16
  • $\begingroup$ I would like to thank you for taking the time to answer my question. $\endgroup$ – user3568 Jun 3 at 12:19
  • $\begingroup$ @Ovi You are right. Thanks. $\endgroup$ – user3568 Jun 3 at 12:20

I would not use either form but employ a formal mathematical expression for $\alpha$ and $\beta$ that more precisely defines them. That avoids English entirely as a weak link in conveying your intent. Use slides or a white/blackboard to be explicit about these things when presenting.

English is full of ambiguity (and I'm a native speaker) and for clarity it is best to resort to mathematics proper when defining things.

It's also useful in presentations to have key definitions visible on e.g. a white board so that anyone can refresh their memory at a glance.

  • $\begingroup$ Thanks for useful notes.I got it. $\endgroup$ – user3568 Jun 3 at 12:13

As native English speaker, both sentences convey the same meaning: here is a field ${\mathbb F}_q$ and consider (the) two elements $\alpha$ and $\beta$ in there. Another poster has said that "the" conveys the intent that there are only two elements: this isn't true because you introduce the elements by noting that there are $q$ elements in the field. It does however suggest that you want to emphasize these two elements for a reason.

"the" is used in mathematical English in the same way as in non-mathematical English -- which is perhaps a little unfortunate as the way English uses articles (the words "a", "an" and "the") is not always easy to understand. However:

  1. "the" can be used to indicate a specific instance of something: "let us consider the unique (up to isomorphism) field of...". Here, we cannot drop "the" because we're talking about an object of some kind, so we need an article to introduce it. We can't use "a(n)" because it's unique, and "a(n)" implies there is more than one

  2. "the" can be used to introduce plural objects: "here are the elements of the ring...". "A(n)" introduces singular objects, or count nouns (e.g. "a crowd of people", "a gaggle of geese")

  3. we can drop the article ("a", "an" or "the") when talking about an abstract quantity, which reinforces that it is abstract: "We have elements that together form a ring" is equivalent to "We have the elements that together form a ring".

In point 3, the word "the" does subtly change the meaning of the sentence, but not grammatically. It adds a little extra emphasis to the noun (here, "elements") that you might want for a specific reason.

  • $\begingroup$ You know when I searched "Consider the two elements" I saw there are 5B results and when I searched "Consider two elements" I saw there are 190k results in google and this confused me. $\endgroup$ – user3568 Jun 3 at 12:19
  • $\begingroup$ @user3568 ghits (google hits) can be very misleading, especially when you have common words like "the" involved. English uses articles in very subtle ways sometimes, and the only real way to be certain is to ask a (good) native speaker how they'd understand it. I think asking here was a better idea than asking google :) $\endgroup$ – postmortes Jun 3 at 12:24
  • $\begingroup$ You are right. I like your answer and your nice comments. Thanks again. $\endgroup$ – user3568 Jun 3 at 12:27

Option A sounds slightly off to me, and I prefer option B. It's a little difficult for me to explain why, but here's my attempt: because $\alpha$ and $\beta$ have not been previously introduced, nor are they in some way uniquely determined.

I looked up the definition of "the", and I found this:

  • denoting one or more people or things already mentioned or assumed to be common knowledge.

  • used to refer to a person, place, or thing that is unique.

Neither of these criteria are satisfied in this case.

  • $\begingroup$ Useful comments. Thanks. $\endgroup$ – user3568 Jun 3 at 12:54

The two cases convey different meanings:

  • In the first case, it sounds like $\mathbb F_q$ has only two elements, and the two elements are $\alpha$ and $\beta$
  • In the second case, $\alpha$ and $\beta$ are two of all possible elements of $\mathbb F_q$ (which may be the same element).
  • 1
    $\begingroup$ As an native English speaker I would not automatically take the first meaning. $\endgroup$ – StephenG Jun 3 at 11:57
  • $\begingroup$ Actually, for $q=2$ the first case would make sense, because $\Bbb F_2$ has only two elements. But I do not believe that this is intended. $\endgroup$ – Dietrich Burde Jun 3 at 11:57
  • $\begingroup$ I agree with you that the wording of the sentence is ambiguous at best. That's what is seems like to me. It could also refer to two specific elements, however I have even harder time imagining that case. $\endgroup$ – Riccardo Sven Risuleo Jun 3 at 12:01

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