# What does “a set of things” mean?

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
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

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.