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Suppose we defined some mathematical object $P$, where $P$ is natural number, polynomial, endofunction, geometric figure, etc. What does the expression “$A$ is a set of $P$s” mean:

  • Set inclusion) For all $a\in A$, $a$ is a $P$.
  • Set equality) For all $a$, $a\in A$ iff $a$ is a $P$.

If both are used, which is the most widespread one (which I can use on the Internet not explaining what I mean)?

Update 0: What does the translation to another language of the expression above mean? (Describe your native language.)

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Regarding the update: If it's a good translation, it will surely mean (approximately) the same thing. Of course, it's possible that something gets lost in translation, so that the translated phrase has a broader or a narrower meaning than the original, but even so, the fundamental point of translation is to map some phrase in one language to the phrase in another language with (in a given context) the most similar meaning possible. –  Ilmari Karonen Aug 27 '11 at 11:24
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When I say "this is a box of cats," is there any reasonable context in which I could mean "this is the box of all cats"? –  Qiaochu Yuan Aug 27 '11 at 16:33
    
@Qiaochu Yuan: I dunno. This is still a question about mathematics, and mathematics does not normally discuss putting cats into boxes. :) –  beroal Aug 29 '11 at 12:18
    
@beroal, Qiaochu: Indeed, that's physics, not math. (However, this question is really more about language than math anyway, even if it does refer specifically to language as used by mathematicians.) –  Ilmari Karonen Aug 30 '11 at 0:17
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1 Answer

I'd say "$A$ is a set of $P$s" for the first, and "$A$ is the set of (all) $P$s" for the second.

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Interesting. However, there may be no “a”/“the” in another language! –  beroal Aug 27 '11 at 10:27
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@beroal: Indeed, but I'd expect any language to have some way of making the distinction between "a collection of some Ps" and "the collection of all Ps". For example, in Finnish (which has no articles like "a"/"the"), I'd translate "A is a set of Ps" as A on joukko P:itä (using the partitive case), while "A is the set of Ps" would be A on P:iden joukko (using the genitive case). –  Ilmari Karonen Aug 27 '11 at 10:32
    
In (some?) other languages without such a distinction, a word meaning "all" is mandatory to represent the second idea. –  Mark S. May 14 '13 at 19:16
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