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When we prove something, we use mathematical symbol ∀ to stand for "for all."

Does it make any difference if we use same symbol for "for any."?

Ben Grossmann
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jessie
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    Do not use symbols to replace words. – Mariano Suárez-Álvarez Feb 01 '16 at 20:43
  • There is no symbol for "for any". Colloquially "for all elephants" = "for any elephant" = "for every elephant". – fleablood Feb 01 '16 at 20:46
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    Yes "for any" and "for all" should be interchangeable. But can you post the specific sentence so we can better evaluate? – Gregory Grant Feb 01 '16 at 20:46
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    The words "for any" tend to be confusing in math because of ambiguity, and should be avoided. (And don't substitute $\forall$ or any symbols for words in a non-formal sentence, as a rule, as Mariano says.) – Thomas Andrews Feb 01 '16 at 20:46
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    For instance, the question "Is it true for any even number $n$ greater than two that $n$ is prime?" does not mean "for all," which is the cause of the ambiguity. It is a phrase best avoided. "For every" or "For all" is good. – Thomas Andrews Feb 01 '16 at 20:48
  • @ThomasAndrews: I would read "Is it true that for any even number $n$ that $n$ is prime?" as meaning the same as "Is it true that for all even numbers $n$ that $n$ is prime?" and $n=4$ would be a counterexample to each. A different statement for me would be "Is there any even number $n$ such that it true that $n$ is prime?" with $n=2$ being an example. – Henry Feb 01 '16 at 20:58
  • @ThomasAndrews that's concept(math) => english not concept(english) to math. If the english concept is clear, than one can use the appropriate symbol regardless of the english word. If the english concept is unclear than it needs to be fixed. But one shouldn't try to code word =>symbol without doing word => cocept => symbol first. ... actually, I'm not sure I see that your except isn't equiv for all. If you were asking "are there any" you wouldn't have phrased it as such. But "any" in are there any is different then "any" in for any. – fleablood Feb 01 '16 at 21:01
  • Wow, case in point, @fleablood. "concept(math)=>english" is not useful at all to explain a point. But thanks for the fine example! – Thomas Andrews Feb 01 '16 at 21:07
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    The fact that you are asking this strongly suggests to me that you shouldn't be using $\forall$ at all, even to mean "for all". By analogy, we know $x^2$ is the square of $x$, so the superscript $^2$ denotes "square of"; but would you write, "In a right triangle, the $^2$ the hypotenuse is equal to the sum of the squares of the other two sides"? – David K Feb 01 '16 at 22:41
  • @Henry: You misread his question, which asks "Is it true for any ... that ..." and not "Is it true that for any ...". His question means "Is there any ... for which it is true that ...", which is certainly not asking about universal quantification. – user21820 Feb 05 '16 at 05:47

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