I thought the | symbol meant "divides by", but in set theory, does it mean something different?

Question

I thought the | symbol meant "divides by", but in set theory it seems that it means "such that." However, I thought we wrote "such that" as :.

Can anybody elaborate?

Answer 1

This shouldn't be surprising. There are only so many letters that we can easily access.

1. $\Bbb{P,Q,R}$ can denote a forcing notion, and have nothing to do with probability, the rationals or the reals.

2. $\pi$ denotes a real number, the ratio between a diameter and circumference of a circle, but in many places it also denotes a projection map, or an automorphism, or just a function.

3. $\Bbb N$ sometimes denote $\{1,2,3,\ldots\}$ and sometimes $\{0,1,2,\ldots\}$.

4. Since set theory is rarely worried with the natural numbers and their arithmetic, there's little point in having a useful symbol like $|$ denoting something like divisibility relation.

5. The same can be said about terminology. Regular and normal objects appear everywhere. Saying that something is weakly compact in set theory is not the same as saying it is weakly compact in functional analysis; a critical point in set theory is not the same as a critical point in real analysis; and so on and so forth.

Different fields have different conventions. The important thing is that (1) you're consistent with your notation throughout a particular text; and (2) the reader knows what you mean when you write mathematical notation.

Answer 2

Any of the following might mean "such that":

$: \ ; \ , \ / \ | \ s.t.$

I don't know if there are others.

Answer 3

Usually, in elementary number theory and the like, the symbol "$\mid$", which is typeset by \mid, means "divides." That is, $2\mid 4$ means "$2$ divides $4$," whereas something like $2\not\mid 3$ would mean "$2$ does not divide $3$." The difference between : and | in set theory is really a nominal one. The symbol | is used in set-builder notation. The linked to page gives a description of this notation in use when describing the set of even integers: $$ \{a\mid a\in\mathbb{Z}\quad\text{and}\quad\exists p\in\mathbb{Z}\;(a=2p)\}. $$ There really is not any important difference so as to invalidate one use of : as opposed to |. As stated, this difference is a nominal one (i.e., "in name only"...or, in this case, "in symbol only").

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Added: As can be seen on this page, in the section explicitly devoted to "such that," common indications of "such that" in mathematics include :, |, and s.t. ... as others have pointed out though, there truly is a bewildering array of symbols and ways people can communicate "such that" in mathematics. I have seen $\ni$ before, as Omnomnomnom points out, and I may have seen one or two other notations, but the important point is this: pick a notation and stick with it. Don't start changing it around. That may confuse readers, and it may lead people to believe that there actually is a difference, when, in fact, there is not.