TITLE: Notation for equipotent sets
QUESTION [1 upvotes]: Question:
What is the difference between $|A| = |B|$ and $A \sim B$ ?
My reasoning:
1) $ $ $|A| = |B|$ means that the power of set $A$ is equal to the power of set $B$
2) $ $ $A \sim B$ means that set $A$ is equipotent to set B
3) Those two notations are equivalent
Objection:
I've got the feeling that in some cases, those two notations will mean something slightly different.

REPLY [3 votes]: The notation $|A|=|B|$ will be widely understood to mean that the two sets have the same cardinality (which is the same as saying they are equipotent). At least when it is clear in the context that $A$ and $B$ are sets rather than something that have a more relevant notion of "size" or "length" applying to them.
$A\sim B$ without further specification cannot be expected to be understood that way. Textbooks often need a single symbol for this because they need to talk about being equipotent before they have developed a scheme for which kind of mathematical object a "cardinality" should be. There are textbooks that choose to use $\sim$ for this, but there is not agreement between different textbooks about which symbol should be used.
Other symbols that are sometimes used to mean "equipotent" are $\cong$ and $\simeq$ (see the comments for even more). But neither these nor $\sim$ always mean "equipotent"; they have a wide range of different meanings across fields of mathematics. In particular $\sim$ is often used as a generic name for any (usually) symmetric relation an author defines for a particular purpose.
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