Kip >i do want to apologize for a duplicate question, i saw too late it had already >been asked. Thanks for the elaboration. Still i do not see in your >elaboration, which seems primarily about enclosing, a justification for the >ordering of sets, while, eg, nub is not? Sorting in sets seems primarily an >algorithmic not mathematical issue.
greg ~krsnadas.org -- from: km <[email protected]> via forums.jsoftware.com to: "[email protected]" <[email protected]> date: 16 November 2013 12:00 subject: Re: [Jprogramming] Powersets (was RE: Sets) >OK, now: but the discussion is mathematical and philosophical and maybe >belongs in Chat. >In math the notations { 0, 1, 2 } , { 1, 2, 0 } , and { 1, 0, 2, 1 } all refer >to THE set whose elements are 0 and 1 and 2 . There is no order in this set, >although there is order in notations naming the set. Also, an element cannot >belong to a set more than once, although the same element may be named more >than once in the notation. Once you know the elements are the numbers 0 and 2 >and 1 you know what set you are talking about. There are things about this set >we do not care about. Is it blue? Is it left-handed? We don't care. The set is >a unique mathematical object, but as is common in math, we don't care even >what the object IS , we only care that it is a set, whatever that is, and that >its elements are the three numbers 2 and 0 and 1. >In a J implementation we DO care what J object is representing a set, and we >would like the J representation to have properties matching the mathematical >properties. So given my definition set =: [: < [: /:~ ~. >and the understanding it is to be applied to a list of boxes, we find that the >notations set 0;1;2 and set 1;2;0 and set 1;0;2;1 all represent the same J >object, namely < 0;1;2 . With my definition < 0;1;2 is THE J object >representing THE mathematical set whose elements are 1 and 2 and 0 . >Why do I have an enclosing box for the set and for each element in the >definitions "a set is a box enclosing a sorted list of boxes", and "an element >is the contents of a box in the sorted list" ? >A fair reason for the outer enclosing box is that in math a list and a set are >different. A mathematical list can have duplicates, a mathematical set cannot. >An order is a required part of a mathematical list, but not of a mathematical >set. Because of the outer enclosing box, my J sets are atoms, not lists. >A good reason (in my opinion) for the outer enclosing box is that with my J >definition the empty set is a normal set, namely set '' , a box enclosing an >empty list. The empty list has no boxes and hence the empty set has no >elements. >A psychological reason is that the outer enclosing box comfortingly resembles >the enclosing braces in the mathematical notation. >Why are elements boxed? That permits any J array to be an element. The >elements of set 4 ; i. 2 2 are the number 4 and the 2 by 2 matrix i. 2 2 . --Kip Murray -- from: greg heil <[email protected]> to: Programming forum <[email protected]> date: 16 November 2013 09:14 subject: Re: [Jprogramming] Powersets (was RE: Sets) Perhaps now? Several times you have asked that revisions including sort be made ... yet my understanding of the classical definition of a set is _unordered_...? greg ~krsnadas.org -- from: km <[email protected]> to: "[email protected]" <[email protected]> date: 16 November 2013 08:57 subject: Re: [Jprogramming] Powersets (was RE: Sets) >...Why do I include Sort /:~ in my proposal? Let's discuss that another time... ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
