Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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:

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

share|cite|improve this question
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
You could write $x \in A$, $x \notin B$. That's just a few symbols more... – TMM Nov 10 '12 at 18:33
up vote 32 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 $B$.

share|cite|improve this answer

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

(For typesetting in LaTeX, for example, on 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: $$x \in A \notin B \not\equiv x \in A \land x \notin B.$$

share|cite|improve this answer
So, you suggested \setminus rather than -? +) – Babak S. Aug 8 '13 at 9:32

Next formula describes your relation

$A \ni x \notin B$

share|cite|improve this answer
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
$\ni$ 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
@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
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.