As far as I know there are two methods of describing sets, Set builder and element tester (If there are known by alternative names please inform)
Both of the approaches operate on one element at a time, either construct one element at a time or test each candidate to see if it satisfies a rule.(Assuming rules are testable on a given variable).
Are there alternatives to the above methods? e.g. describing a set by relationship of each element to all other elements, a possible example of such behavior might be : Given n points on plane, set of all circles where total distance of all points to each circle is minimized. Is there an example of a set described by intrinsic dependency between the elements of it's members such that having $n-1$ of it's elements correct or none of it's elements correct to be indistinguishable?
If you have a property $P$ and you can prove that there is a unique set $s$ such that $P(s)$ holds, then you've defined $s.$ You may, however, have no way of determining whether something is a member of $s$ or not.
Here's a concrete example from standard mathematics:
First let $Q(x)$ be the following property:
\begin{align}&x\text{ is a collection of sets of real numbers,} \\&\quad\text{every open set of real numbers belongs to }x, \\&\quad\text{and }x\text{ is closed under the operations of complementation and countable union.} \end{align}
Now define the property $P(x)$ to mean: $$Q(x)\text{ and }(\forall y)(Q(y)\implies x\subseteq y).$$
Then there is a unique set $\scr{B}$ such that $P(\scr{B})$ holds; $\scr{B}$ is called the collection of Borel sets. But this is nothing like a definition of $\scr{B}$ in set-builder form or the like; there's no easy way to take a given set $X$ of reals and test to see if $X$ belongs to $\scr{B}$ or not.
(In fact, one can come up with an alternative definition of $\scr{B}$ that's more constructive than this, but it involves a transfinite induction of length $\aleph_1.$)