Hi Steven, I think you are taking this a bit too far out of what we normal use Python for in real life :-)
The mathematical complication of not having ∀x(x∈S ↔ x∈T) → S = T be a consequence of S = T → (∀x)(x∈S ↔ x∈T) which may sound weird to many people, originates in the fact that the above must also hold for infinite number of elements in a set, including uncountably infinite sets and sets which have such sets as elements (including possibly uncountably infinite recursion of such inclusions). You enter a world full of wonders when you start considering such things. In Python, however, we typically always operate on finite sets. In such a world, the above is not a complication anymore. In fact, the C implementation uses: |S| = |T| ∧ S ⊆ T → S = T or written using the above form: |S| = |T| ∧ ∀x(x∈S → x∈T)→ S = T Cheers. On 3/24/2020 11:19 AM, Steven D'Aprano wrote: > Apologies in advance... the following is going to contain mathematical > jargon and pedantry. Run now while you still can *wink* > > > On Tue, Mar 24, 2020 at 12:56:55AM -0700, Andrew Barnert wrote: >>>>> On Mar 23, 2020, at 19:52, Steven D'Aprano <st...@pearwood.info> wrote: >>>> On Mon, Mar 23, 2020 at 06:03:06PM -0700, Andrew Barnert wrote: >>>> The existing methods are named issubset and issuperset (and >>>> isdisjoint, which doesn’t have an operator near-equivalent). Given >>>> that, would you still want equals instead of isequal or something? >>> Oops! I mean, aha, you passed my test to see if you were paying >>> attention, well done! >>> >>> *wink* >>> >>> I would be satisfied by "isequal", if the method were needed. >>> >>> >>>> So, I’d rather have an uglier, more explicit, and more obviously >>>> specific-to-set name like iscoextensive. Sure, not everyone will know >>>> what “coextensive” means >>> That's an unnecessary use of jargon >> As far as I’m aware (and your MathWorld search bears this out), >> “coextensive” isn’t mathematical jargon. > No, it's definitely mathematical jargon, but *really* specialised. As > far as I can tell, it's only used by people working on the foundations > of logic and set theory. I haven't come across it elsewhere. > > For example: > > https://books.google.com.au/books?id=wEa9DwAAQBAJ&pg=PA284&lpg=PA284&dq=math+coextensive+set+difference+with+equality&source=bl&ots=y-jmrSTxC8&sig=ACfU3U18oCjeXSfwwqML1zuVWwbSJdeZ2Q&hl=en&sa=X&ved=2ahUKEwjqzfGBhbLoAhUOzDgGHfqSAbIQ6AEwAHoECAkQAQ#v=onepage&q=math%20coextensive%20set%20difference%20with%20equality&f=false > > "Introduction To Discrete Mathematics Via Logic And Proof" by Calvin > Jongsma, on pages 283-4, it says: > > First-Order Logic already has a fixed, standard notation of > identity governed by rules of inference. S = T logically implies > (∀x)(x∈S ↔ x∈T) [...] Thus identical sets must contain exactly > the same elements. However we can't turn this claim around and > say that sets having the same elements are identical -- > *set equality for coextensive sets doesn't follow from logic alone.* > We'll therefore postulate this as an axiom, using the formal > notation of FOL. > > Axiom 5.3.1: Axiom of Extensionality > ∀x(x∈S ↔ x∈T) → S = T > > The crierion for equal sets (see Definition 4.1.1 follows > immediately: > > Proposition 5.3.1: Equal Sets > S = T ↔ ∀x(x∈S ↔ x∈T) > > > Apologies to everyone who got lost reading that :-) > > In the above quote, "identity" should be read as "equality". The > emphasised clause "set equality for coextensive sets..." is in the > original. The upside down A ∀ means "for all", the rounded E ∈ means > "element of". > > So translated into English, what the author is saying is that he has a > concept of two sets being equal, S = T. He has another concept of two > sets being coextensive, namely, that for each element in S, it is also > in T, and vice versa. He then says that if two sets are identical > (equal), then logically they must also be coextensive, but to go the > other way (coextensive implies equality) we have to make it part of the > definition. > > That's a long-winded, pedantic and precise way of saying that two sets > are equal if, and only if, they have precisely the same elements as each > other, by definition. > > >> I chose it because of its ordinary (if not super-common) English >> meaning. > Oh. > > Well, you outsmarted me, or perhaps outdumbed me *smiles* because it > definitely is a term from mathematics, whether you knew it or not. > > I thought that you were referring to the mathematical usage. > > As far as the non-mathematical meaning: > > being of equal extent or scope or duration; > having the same spatial limits or boundaries; > > I think we would be justified as reading `A.iscoextensive(B)` in one of > two ways: > > len(A) == len(B) > (min(A), max(A)) == (min(B), max(B)) > > but not necessarily as equal. > > Why not equal? Consider somebody who started a project on the day of the > Olypics Opening Ceremony, and finished the project on the day of the > Closing Ceremony. We can say that the project and the Olympics were > coextensive, but we can't necessarily say that they were equal or the > same thing, or that the project was the Olympics. > > > [...] >> Sure, but in math, just as in Python, a set is never equal to a list >> (with a tiny foundations asterisk that isn’t relevant, but I’ll >> mention it below). > Well, yes and no. > > In Python, just as in mathematics, a set is never a superset or subset > of a list either, and yet in Python we have a set method which quite > happily says that a set can be a superset or subset (or equal to) a > list: > > {1, 2}.issubset([2, 1]) # Returns True > > We understand that, unlike the operators `<` and `<=`, the issubset > method is happy to (conceptually) coerce the list into a set. > > >> So when mathematicians want to say a set is equal to a list… well, >> they just don’t say that, because it isn’t true, and usually isn’t >> even a meaningful question in the first place. > Don't you think that there is a sense in which the Natural numbers ℕ {0, > 1, 2, 3, ...} and the sequence (0, 1, 2, 3, ...) are "the same thing"? > > We even talk about ℕ having successor and predecessor functions, which > implies a natural order to the set. > > >>> The foundations of mathematics is one of those things which are >>> extremely over-represented on the Internet >> Well, you’re the only one who brought up foundations here, and in the >> same email where you’re railing against people talking about >> foundations on the internet, so I’m not sure what the point is. > Because I genuinely thought you intended to refer to the mathematical > jargon, I didn't imagine you just got lucky. > > >> My guess is that you saw Greg, Marc-Andre, etc. talking about sets in >> mathematics and naturally thought “oh no, here comes irrelevant >> mathematical logic”, > *blink* > > Um, are you confusing me with someone else? Is there something I've said > that leads you to the conclusion that I think that being mathematical > and logical is irrelevant? > > >> but there isn’t any; the reason sets came up is >> that we’re talking about set operations on Python set objects, and >> it’s pretty hard to talk about what issubset means/should mean without >> talking about sets. > Oh well I'm glad you set me straight. > > -- Marc-Andre Lemburg eGenix.com Professional Python Services directly from the Experts (#1, Mar 22 2020) >>> Python Projects, Coaching and Support ... https://www.egenix.com/ >>> Python Product Development ... https://consulting.egenix.com/ ________________________________________________________________________ ::: We implement business ideas - efficiently in both time and costs ::: eGenix.com Software, Skills and Services GmbH Pastor-Loeh-Str.48 D-40764 Langenfeld, Germany. CEO Dipl.-Math. 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