On Mon, Aug 10, 2020 at 2:20 PM Stephen J. Turnbull < email@example.com> wrote:
Could we add the idea of "negative" sets to Python? That means sets that contain EVERYTHING EXCEPT certain elements.
This is usually called the "complement" of a set. Since (in set theory) there is no set of all sets, "absolute" complement is an unfounded concept. The idea of having a universe and defining "absolute complement" as complement relative to the universe is often adopted, but it has set-theoretic problems (the universe doesn't have a powerset, for one thing), and frequently you end up with a hierarchy of universes (categorists who try build category theory on set theory run into that a lot).
Yes, we can implement things in Python that aren't allowed in formal mathematics and some fun questions arise. What should `set.UNIVERSAL in set.UNIVERSAL` return? Bertrand Russell thinks it's False.
Considering those points, this proposal seems very abstract. I think it's fun to think about, and maybe it has practical applications in constructing other sets. But a "set" that isn't iterable, and whose "in" is logically equivalent to "not in" its complement (which is a Python set!), doesn't seem directly useful in itself.
I think it'd be best if such a class was not a subclass of the Set ABC, since it's not iterable. You can use it like a set in certain situations where you know what you're doing, but avoid passing it to generic code that expects a real set.
I think this is one where you need to present both use cases and an implementation. Speaking of implementations and fun:
First, let's have a universal set that contains everything.
assert element in set.UNIVERSAL
For what values of "everything"? Python sets cannot contain all the objects of Python. Specifically, an element of a set must be hashable. Will that be true for your universal set?
The builtin set requires hashability, but you could certainly implement a subclass of the Set ABC that satisfies all contracts and accepts non-hashable elements. It would just be much less performant.
However REMOVING an element from the set puts it on "negative list".
myset = set.UNIVERSAL
Shouldn't this be "myset = copy(set.UNIVERSAL)"?
You'd like it to be "myset = set(set.UNIVERSAL)", I guess, and that indeed would require adding set.UNIVERSAL to Python.
I think the better API would be `myset = UniversalSet()`.
Anyway, in the end, I think we all agree that this is fun to think about but too obscure for the Python language itself. Maybe you could put it on PyPI.