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Suppose I wanted to say that

$$x \in A \notin B$$.

Is there a (better) standard way to describe this? Else, I'll go for my original formulation:

$$ \ldots \text{where}\, x \in A\,\text{ but not in } B$$

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3  
Writing it in words is almost always so much clearer... To make it even better, try "where $x$ is in $A$ but not in $B$". –  lhf Nov 10 '12 at 18:29
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You could write $x \in A$, $x \notin B$. That's just a few symbols more... –  TMM Nov 10 '12 at 18:33
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3 Answers

up vote 30 down vote accepted

What you want is $x\in A\setminus B$: the set $A\setminus B$ is by definition the set of things that are in $A$ but not in $B$. (An older notation is $A-B$; I don’t recommend it.)

The expression $x\in A\notin B$ says something entirely different: it says that $x$ is an element of $A$, and $A$ is not an element of $B$.

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Your statement can be written with the set-minus character: $\setminus$

(For typesetting in LaTeX, for example, on math.se: use \setminus):

$$x \in A\setminus B,$$ which is defined to be exactly:

$$x \in A \land x \notin B$$

While you can chain together set inclusion $\subset$, e.g. $x \in A \subset B \subset C$ from which it follows that $x \in A \land x\in B \land x\in C$, that's not appropriate for set membership: $$x \in A \notin B \not\equiv x \in A \land x \notin B.$$

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So, you suggested \setminus rather than -? +) –  B. S. Aug 8 '13 at 9:32
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Next formula describes your relation

$A \ni x \notin B$

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Some might argue that you are prefacing a statement about A, with asserting "A such that x is not in B" which might be confusing. There was just yesterday a "debate" about the notation $\ni$ and its signification: "such that..." vs. reversed set-inclusion (the correlate of $\supset$), etc. Anyway, +1 for the creativity! –  amWhy Nov 10 '12 at 21:32
    
Just look at the LaTeX notation to clarify the meaning of $\ni$. Wiki –  Sergei Nov 10 '12 at 21:50
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$\ni$ does not appear to be in that table. Even if it were, it is still a confusing notation since many people use $\ni$ and/or $\backepsilon$ to mean "such that". A more common notation would be $x\in A\setminus B$ or $x\in A-B$ –  robjohn Nov 10 '12 at 22:21
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@robjohn Wiki was about "such that". It may be expressed as ":" or "|". If some people use $\ni$ as "such that" they do it wrong. There is nothing in use two relations in one expression instead of two expressions. Like $1<x<2$. But yes, $x \in A \backslash B$ is more clear to understand. –  Sergei Nov 10 '12 at 22:32
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Potentially, $\ni$ for reverse inclusion could be useful. For instance, if you want to sum over all sets $A$ containing $x$, you could write $\sum_{A \ni x} f(A)$. But I have to agree that $A \ni x \notin B$ is ugly... –  TMM Nov 11 '12 at 1:31
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