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As someone not very proficient in set theory (and am currently working on a proof in another area), I was unsure when attempting to define a set consisting of some elements satisfying a particular condition. As a very basic example, how would I define a set containing some arbitrary number of even numbers? Is there something other than the following:

$$S := \{\text{some } x \in \mathbb{R} : \text{x is even}\}$$

The idea is that $S$ contains some arbitrary number of even numbers (could just be $\{2\}$, or {2, 6, 8}, or all even numbers, etc.). Moreover, what would be the proper notation for ensuring that this set contains at least one element? In a similar vein, how would one define a set $A$, for example, as a subset of $S$, for which at least one element not satisfying some condition is removed (if there exists an element in S at all which can be removed)?

Until now I've only worked with sets with some absolute condition (all elements satisfying a condition, etc.) and am lost as to how to construct sets as described above.

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  • $\begingroup$ I can't understand what set $S$ you want. Usually, it suffices to give a description of $S$ in English like "Let $M(\mathbb{R}^n)$ be the set of complex Borel measures on $\mathbb{R}^n$". $\endgroup$
    – Mason
    Commented Feb 10, 2022 at 22:04
  • $\begingroup$ @Mason I apologize that my question is not clear. I am attempting to define a recursive sequence of sets in which some arbitrary number (not all, but at least one) elements are eliminated each time based on some condition. This got me thinking about how I would define a set using the idea of "some arbitrary number" in the first place, so wanted to start with a simple case. $\endgroup$
    – tmako
    Commented Feb 10, 2022 at 22:07
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    $\begingroup$ Why not just use $\emptyset\neq S\subseteq\{x\in\mathbb{R}:x\text{ is even}\}$? You aren't specifying what $S$ is uniquely so you can't really give a definition of the form $S=$ [something]. If you want to specify that something was removed, use $\subsetneq$ instead of $\subseteq$. $\endgroup$ Commented Feb 11, 2022 at 10:59

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$$S:=\{x∈R:x \text{ is even }\}$$

$S:=\{\exists x∈R:x \text{ is even }\}$ or $|S| > 0$

$|S|$ is the number of elements or the size of $S$

Regarding the subset, similar, you can define the subset that you want and speak about the size of subset or difference.

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