Notation for set excluding element

Consider the set $\mathbb{A}=\{a,b,c\}$.

I want to refer to "the set $\mathbb{A}$ excluding the element $a$"

Is the notation $\mathbb{A} \sim \{a\}$ equivalent to $\mathbb{A} \setminus \{a\}$?

Is the former an abuse of logic notation?

The latter is what I am used to.

• Was it possibly hand-written? Because then it could have been a sloppily written $A-\{a\}$. Sep 29, 2016 at 19:07
• @celtschk, good point. Yes it was an informal remark (but typed, not handwritten). But from someone who is usually very careful! I've edited the question since I am still interested in knowing if this is generally seen as 'sloppy' Sep 29, 2016 at 19:09
• Usually not; see Complement. In set theory, the symbol $\sim$ is already used for Equinumerosity. Sep 29, 2016 at 19:13
• Also note that "the set $\mathbb{A}$ excluding the element $\{a\}$" would mean $\mathbb{A}\backslash\{\{a\}\}$. What you probably mean is "the set $\mathbb{A}$ excluding the element $a$"
– cdwe
Sep 29, 2016 at 19:18
• @ cdwe yes. I mean element. fixed the question Sep 29, 2016 at 19:19

The notation $\Bbb A - \{a\}$ is often used to mean the same thing as $\Bbb A \setminus \{a\}$ (the set difference), but I've never seen it with a tilde and can't find any references to it being used this way with Google.
The tilde $\sim$ is sometimes used as a negation or "not" symbol in set theory, in which case
$$\Bbb A \setminus \{a\} = \bigl\{x : x \in \Bbb A, \sim\!(x\in\{a\})\bigr\}.$$
The tilde is also used sometimes for equivalence relations, where $x \sim y$ means $x$ and $y$ are equivalent (in the same equivalence class) under some equivalence relation $\sim$.
A particularly common example of this is with the cardinality of sets. We say $A \sim B$ if $A$ and $B$ have the same cardinality, that is $|A| = |B|$, and we call them equinumerous.
• Thanks – makes all sense. @egreg, do you have any reference where $A\sim B$ is used to denote $A \setminus B$? Sep 29, 2016 at 20:28