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.)
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.