1
$\begingroup$

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.)

  • 0
    @beroal, Qiaochu: Indeed, [that's physics](http://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat), 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.)2011-08-30

2 Answers 2

8

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.

  • 0
    In (some?) other languages without such a distinction, a word meaning "all" is mandatory to represent the second idea.2013-05-14
0

If you want set inclusion, you should say A is a set of P's.

If you want set equality, you should say A is the set of all P's.

The words "a" and "the" (most often) have very specific meanings in mathematics.