Marshall's verb issub needs work. Consider in his model
]A =: (0;1);2 NB. conventional { {0,1} , 2 }
┌─────┬─┐
│┌─┬─┐│2│
││0│1││ │
│└─┴─┘│ │
└─────┴─┘
]B =: (1;0);2;3 NB. conventional { {1,0) , 2 , 3 }
┌─────┬─┬─┐
│┌─┬─┐│2│3│
││1│0││ │ │
│└─┴─┘│ │ │
└─────┴─┴─┘
issub=: *./@:e.
A issub B NB. { {0,1) , 2 } = { {1,0} , 2 } IS a subset of { {1,0) , 2, 3
}
0
Kip
On 10/11/2010 4:53 PM, Marshall Lochbaum wrote:
> Corrections:
>
> set=: [: , boxopen"0
> isset=: -: set
>
> Marshall
>
> -----Original Message-----
> From: Marshall Lochbaum [mailto:[email protected]]
> Sent: Monday, October 11, 2010 5:21 PM
> To: 'Programming forum'
> Subject: RE: [Jprogramming] Sets
>
> I am going to offer a competing model, which will hopefully clarify or
> improve some things.
>
> In this model, a set is simply a list of boxes with no other requirement.
> Sets match if each is a subset of the other, and one set is a subset of
> another if all of its elements are contained in that set.
> Repeating elements is only a dumb way to write something, and can be
> rectified with the verb stdform.
> Cross product is supported with an arbitrary number of sets--see below.
>
> set=: [: ,<"0
> empty=: 0$a:
>
> isboxed=: 32 = 3!:0
> islist=: 1...@$
> isset=: isboxed *. islist
> stdform=: /:~...@~.
> isstdform=: -: stdform
>
> ismemb=: boxo...@[ e. ]
> issub=: *./@:e.
> equal=: issub *. issub~
>
> sor=: ,
> sand=: [ #~ e.
> sdif=: -.
> ssymdif=: sor sdif sand
> card=: #...@~.
>
> NB. monad: accepts a list of sets in boxes. Take the cross product.
> NB. dyad: find the cross product of sets x and y.
> cross=: [: ,@{ ,&(] :<)
> power=:<@#~ #:@i.@(2^#)
>
>
> Marshall
>
> -----Original Message-----
> From: [email protected]
> [mailto:[email protected]] On Behalf Of Kip Murray
> Sent: Sunday, October 10, 2010 12:02 AM
> To: Programming forum
> Subject: [Jprogramming] Sets
>
> Here is my latest attempt to model set theory in J. All sets have distinct
> elements and are ordered by /:~ so that match -: determines whether two sets
> are the same. Sets must be created by the verb set or by provided
> operations. The intention is theoretical not practical! --Kip Murray
>
> ]A =: set 0;'b';2 NB. elements 0 'b' 2 are put in boxes preceding the
> last ┌─┬─┬─┬┐ │0│2│b││ └─┴─┴─┴┘
> ]B =: set 2;'b';'b';0 NB. same elements so same set ┌─┬─┬─┬┐ │0│2│b││
> └─┴─┴─┴┘
>
> A -: B
> 1
>
> ]C =: set 2;'b';'c';'d'
> ┌─┬─┬─┬─┬┐
> │2│b│c│d││
> └─┴─┴─┴─┴┘
>
> B sand C NB. intersection, "set and"
> ┌─┬─┬┐
> │2│b││
> └─┴─┴┘
> B sor C NB. union
> ┌─┬─┬─┬─┬─┬┐
> │0│2│b│c│d││
> └─┴─┴─┴─┴─┴┘
>
> (A sand B sor C) -: (A sand B) sor (A sand C) NB. distributive law
> 1
>
> pwrset A NB. A has 3 elements, power set has 2^3, including the empty
> set ┌──────┬────────┬────┬──────┬────┬──────┬──┬────┬┐
> │┌─┬─┬┐│┌─┬─┬─┬┐│┌─┬┐│┌─┬─┬┐│┌─┬┐│┌─┬─┬┐│┌┐│┌─┬┐││
> ││0│2││││0│2│b││││0││││0│b││││2││││2│b│││││││b││││
> │└─┴─┴┘│└─┴─┴─┴┘│└─┴┘│└─┴─┴┘│└─┴┘│└─┴─┴┘│└┘│└─┴┘││
> └──────┴────────┴────┴──────┴────┴──────┴──┴────┴┘
> NB. Elements are contained in boxes preceding the last which is always
> NB. the Boxed Empty a: (Ace). The use of a: permits a unique and visible
> NB. empty set, viz
>
> (,a:) -: E =: A less A NB. see verb less below
> 1
> E
> ┌┐
> ││
> └┘
> a:
> ┌┐
> ││
> └┘
> E -: a:
> 0
>
> NB. Definitions
>
> E =: ,a: NB. empty set
> set =: a: ,~ [: /:~ ~. NB. create set from boxed list y
> NB. each box of y encloses an element
> get =: { }: NB. get boxed elements (from curtail because
> NB. elements are inside boxes of curtail)
> isin =: e. }: NB. Do boxes in list x contain elements of y?
> less =: a: ,~ -.&}: NB. remove elements of y from x
> sand =: [ less less NB. intersection, "set and"
> sor =: a: ,~ [: /:~ [: ~. ,&}: NB. union, "set or"
> diff =: less sor less~ NB. symmetric difference
> card =: [: # }: NB. count elements: cardinality
> issubs =: [ -: sand NB. Is x a subset of y?
> pwrset =: a: ,~ [: /:~ ] (<@#~) 1 (,~"1) 2 (#"1~ {:)@#:@i...@^ #...@}:
> NB. pwrset by Raul Miller, adapted
> islist =: 1 = #...@$ NB. islist through isunique from validate.ijs
> isboxed =: 0< L.
> issorted =: -: /:~
> isunique =: -: ~.
> isset =: islist *. isboxed *. (a: -: {:) *. issorted@:}: *. isunique@:}:
> NB. isset y asks, is array y a set?
> iselement =:<@[ isin ] NB. Is array x an element of set y?
>
> NB. End
>
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