# Set Notation: How to denote ALL points in some space satisfying a condition?

I am integrating a function over the multidimensional domain $\Omega$, which is a subset of a larger domain $X$. Omega is defined by ALL points $x \in X$ that satisfy some condition, the details of which are not important, that I will denote $\text{cond}\!\left(x\right) = \text{TRUE}$. There are many points throughout $X$ that satisfy this condition (i.e. $\Omega$ is the union of a number of disjoint regions in $X$) due to symmetries. I am having some trouble providing an unambiguous definition of $\Omega$ in set notation. Here is what I've got:

$$\Omega = \left\{x \mid x \in X, \text{cond}\!\left(x\right) = \text{TRUE}\right\}$$

Does this clearly imply ALL points $x \in X$ that satisfy the condition (i.e. even the symmetrically equivalent points--they need to be included in the integration even though it is essentially double counting)? Or is there a better way to make this clear?

• This looks fine to me. You could even shorten it to $$\Omega = \{x\in X | \operatorname{cond}(x) =\textrm{TRUE}\}$$ – MPW Feb 1 '16 at 17:49
• Yes, that's what this notation means. It doesn't take into account symmetries in any way. – Qiaochu Yuan Feb 1 '16 at 17:49
• @MPW: If you submit your answer as an answer I would be happy to accept it. Thank you. – okj Feb 4 '16 at 1:02

## 1 Answer

[Converted from comment to answer]

This looks fine to me. You could even shorten it to $$\Omega = \{x\in X \mid \operatorname{cond}(x) =\textrm{TRUE}\}$$

Addendum: Since the condition is evidently truth-valued, you could even write just

$$\Omega = \{x\in X \mid \operatorname{cond}(x)\}$$