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.