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There are lots of expressions like, $\text{for all}\ x$, $\text{for any}\ x$, $\text{for some}\ x$, etc.

It may be because of the fact that I am not a native English speaker, but I can't tell the difference between "for all" and "for any."

(I think "$\text{for some}\ x$" means 'there exists at least one point', right?)

Could you explain it? Thank you!

RyRy the Fly Guy
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syko
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    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. – syko May 25 '13 at 13:23
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    I agree that the word "any" is sometimes confusing. I almost always write "for each". I think it's much clearer. – Carl Offner May 25 '13 at 15:35
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    This answer http://english.stackexchange.com/a/50951/10447 over on English seems helpful to me. – David E Speyer Dec 20 '13 at 16:27

3 Answers3

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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.

KCd
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    You are the best! – syko May 26 '13 at 02:18
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    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' ? – lucidbrot Jan 10 '18 at 07:20
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    What do you think? – KCd Jan 10 '18 at 13:41
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    This "any" term usage made me take lot of time to understand the meaning of Riemann integral definition, given in Bartle and Sherbert textbook. – Sensebe Oct 21 '18 at 07:20
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    @lucidbrot "Does that mean that in mathematical context, it always holds that for any ≠ for some?" $\quad$ Unfortunately, no, not even if we consider just mathematical statements (rather than questions); adding to KCd's answer: (1) when for any is embedded in a conditional's antecedent it occasionally means for some, and (2) not any does not mean not all. Elaboration. – ryang Jul 19 '23 at 03:16
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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 $x=x$".

Easy
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    When giving a downvote, please provide with your reason, otherwise no one would where the problem is. – Easy May 25 '13 at 13:30
  • Thank you for your answer. It helps me a lot – syko May 25 '13 at 13:51
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    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. – Peter Smith May 25 '13 at 13:51
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    @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. :) – 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".

kinbiko
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    Thank you for your answer and explanation. It is helpful – syko May 25 '13 at 13:51
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    Do I understand your answer correctly, that you're saying that for any is equivalent to for an arbitrary? – lucidbrot Jan 10 '18 at 07:23
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    @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 – kinbiko Jan 10 '18 at 12:01
  • @kinbiko How "for some" should interpreted it as "at least"? With "at least" if it is true "for all" then "at least" is also true. But "some" means "a part but not all" so it would be false. – Antonios Sarikas Jul 27 '20 at 09:53