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There are lots of expressions like, for all x, for any x, for some x, etc.

I think 'for some x in R s.t ~' means that there exists at least one point in R s.t ~~. right?

However, I can't know the difference between 'all' and 'any'.

(It may be because of the fact that I am not a native english speaker.)

Could you explain it? thank you~!

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  • $\begingroup$ I read a sentece, "Suppose f: A in R^n -> R is differentiable on an open set A. For any x, y in A such that the line segment joining x and y lies in A ( which need not happen for all x,y ), there is a point c on that segment such that f(y)-f(x)=Df(c)(y-x)." This is the sentence that confused me. However in the past I already had some curiosity about it. $\endgroup$ – syko May 25 '13 at 13:23
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    $\begingroup$ I agree that the word "any" is sometimes confusing. I almost always write "for each". I think it's much clearer. $\endgroup$ – Carl Offner May 25 '13 at 15:35
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    $\begingroup$ This answer english.stackexchange.com/a/50951/10447 over on English seems helpful to me. $\endgroup$ – David E Speyer Dec 20 '13 at 16:27
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The term "any" is troublesome, because in natural usage it could mean "all" or "at least one", depending on the context. Here are examples to consider.

(1) For any $a > 0$ there is an $x > 0$ such that $x^2 = a$.

(2) Does the equation $x^3 + y^3 + z^3 = 33$ have any integral solution?

(3) Have you solved any of those problems?

(4) Using this new technique, I can solve any of the problems from that list.

In the first example, "any" = "all". In the second one, "have any" is asking about existence. In the third, "any" means "at least one" (existence). In the fourth, "any" means "all".

I have known weak math students who are native English speakers and think (1) is proved by showing it works when $a = 1$, even though that way of interpreting (1) makes it into a trivial statement. In other words, they interpret "For any" in (1) as meaning "For some", and hence turn (1) into an existence claim instead of a universal claim. Such usage of "any" is present in non-mathematical English (see the third example), and I think this is the basis for the student's misunderstanding (comparable to having to learn the different meaning of "or" in mathematical English compared to non-technical English). I don't think any native English speaker would misunderstand the different senses of "any" in (3) and (4).

I would advise someone who is not a native English speaker to avoid using "any" in mathematical statements. You can convey what you need with other choices of words.

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  • $\begingroup$ You are the best! $\endgroup$ – syko May 26 '13 at 2:18
  • $\begingroup$ Does that mean that in mathematical context, it always holds that for any ≠ for some ? E.g. in my CS textbook "We store x if u minimizes abs(x-u) for any pair x,u". Does 'any pair' here mean 'all pairs' or 'at least one pair' ? $\endgroup$ – lucidbrot Jan 10 '18 at 7:20
  • $\begingroup$ What do you think? $\endgroup$ – KCd Jan 10 '18 at 13:41
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    $\begingroup$ This "any" term usage made me take lot of time to understand the meaning of Riemann integral definition, given in Bartle and Sherbert textbook. $\endgroup$ – Immortal Player Oct 21 '18 at 7:20
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I just want to point out the difference between "for all" and "for any":

1) "for all" usually used in the end of the sentence, meaning the condition is always satisfied. For example, "$x=x$ for all $x\in\mathbb{R}$".

2) "for any" usually is placed in the beginning of the sentence, stressing that you are choosing an arbitrary element. For example, "for any $x\in\mathbb{R}$, we have $x=x$".

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    $\begingroup$ When giving a downvote, please provide with your reason, otherwise no one would where the problem is. $\endgroup$ – Easy May 25 '13 at 13:30
  • $\begingroup$ Thank you for your answer. It helps me a lot $\endgroup$ – syko May 25 '13 at 13:51
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    $\begingroup$ The claim about beginning of sentence vs end of sentence use doesn't strike me as right (though I haven't done a statistical count of usage in maths books -- have you?). But the distinction you suggest (between saying a condition is always satisfied and saying it is satisfied by some arbitrarily chosen object) is the one suggested by Bertrand Russell in *The Principles of Mathematics", §61. $\endgroup$ – Peter Smith May 25 '13 at 13:51
  • $\begingroup$ @PeterSmith, yeah, the description here is pretty subtle. I was trying to say the beginning vs end as for the "maybe-natrual" order in speaking. But it seems confusing also, so I might actually just want to say the latter meaning that u suggest. :) $\endgroup$ – Easy May 25 '13 at 14:02
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Your interpretation of "for some" is correct.

There is no logical difference between "for all" and "for any", they both mean $\forall$. For any number n in the integers, there is always a number bigger than it. For all numbers in the integers, there is always a number bigger than it.

The difference language-wise is that your sentence must stick to the plural form of nouns when using "for all", while you may use singular nouns when using "for any".

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  • $\begingroup$ Thank you for your answer and explanation. It is helpful $\endgroup$ – syko May 25 '13 at 13:51
  • $\begingroup$ Do I understand your answer correctly, that you're saying that for any is equivalent to for an arbitrary? $\endgroup$ – lucidbrot Jan 10 '18 at 7:23
  • $\begingroup$ @lucidbrot Yes. As, by saying for an arbitrary you are not restricting which element which you're talking about. I.e. it could be any. for an( arbitrar)y $\endgroup$ – kinbiko Jan 10 '18 at 12:01

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