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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.

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    $\begingroup$ Was it possibly hand-written? Because then it could have been a sloppily written $A-\{a\}$. $\endgroup$
    – celtschk
    Sep 29, 2016 at 19:07
  • $\begingroup$ @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' $\endgroup$
    – OO_SE
    Sep 29, 2016 at 19:09
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    $\begingroup$ Usually not; see Complement. In set theory, the symbol $\sim$ is already used for Equinumerosity. $\endgroup$ Sep 29, 2016 at 19:13
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    $\begingroup$ 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 $a$" $\endgroup$
    – cdwe
    Sep 29, 2016 at 19:18
  • $\begingroup$ @ cdwe yes. I mean element. fixed the question $\endgroup$
    – OO_SE
    Sep 29, 2016 at 19:19

1 Answer 1

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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.

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