for the letter Ø , used to signify the empty set, see http://en.wikipedia.org/wiki/%C3%98#Mathematics
--- Den tirs 4/8/09 skrev Kip Murray <[email protected]>: > Fra: Kip Murray <[email protected]> > Emne: Re: [Jprogramming] J Sets > Til: "Programming forum" <[email protected]> > Dato: tirsdag 4. august 2009 05.28 > Raul, about phi, remember the giant > in Jack and the Beanstalk? When he said > Phee, Pheye, Phoh, Phum, he was trying to recite Greek > letters! The symbol Ø > for the empy set looks vaguely like the Greek letter phi Ф > , hence the name phi > for the empty set. You occasionally see the Greek > capital phi used for the > empty set. > > About the catechism, I apologize for not making clear its > title should be > > Catechism for Implementors > > so Question 1 amounts to "how are you representing sets", > just as you knew was > proper. > > The questions where I supply answers give standard > theoretical properties I > would like any implementation to have. I may have to > settle for more than one > empty set, though, as J has empties of different shapes, > and "shape boxing > elements" is what Match -: uses to determine whether two > arrays "are the same". > I think I avoided this problem > with my "set-marker" model, but I am leaning > more now to the broader model of Dan and Fraser. > > Kip > > > Raul Miller wrote: > > On Sun, Aug 2, 2009 at 7:23 PM, Kip Murray<[email protected]> > wrote: > >> 1. Question: What is a set? > >> Answer: > > > > This question is too abstract, I think. A proper > question > > would be "how are we representing sets", and this can > > depend on your application and on your universe of > supported > > elements. > > > > For example, I have no problem with "a set is > represented > > as a sorted list of unique boxes". But I also > have no problem > > with "a set is represented as a sequence of bits > marking > > which element in the universe is a member" > > > >> 2. Question: What is an element of a set? > >> Answer: > > > > See above. > > > >> 3. Question: When are two sets the same set? > >> Answer: Set H is set K provided each > element of H is an element of K and > >> each > element of K is an element of H. > > > > In the context of computer programs, this depends on > how > > you represent sets. For both of my > examples for question 1, > > -: would work. However, -: will not work for all > possible > > representations of sets. > > > >> 4. Question: What is a subset of a set? > >> Answer: To say H is a subset of set K > means H is a set, and each element of H > >> is an > element of K. > > > > Also, the details of how this is implemented can > depend on how > > you represent sets. One fundamental approach > > would be to find the elements in the potential subset > > which do not appear in the potential superset -- if > there > > are none the potential subset really is a subset. > > > > But if you represent sets with equal-length lists of > bits > > you can use an expression which would not work for > > other potential approaches: > > 0 0 1 *./ .<: 0 1 1 > > 1 > > > >> 5. Question: What is an empty set? > >> Answer: The empty set, called phi, is > the set which has no element. > > > > yes. > > > > But I never call the empty set "phi", and this name > > does not seem to be a common name for the empty > > set. Then again, I never call the empty set > "george". > > > >> 7. Question: Why is the empty set a subset of > every set? > >> Answer: > > > > This comes from the definition of subset (and of > set): > > > > The set containing no elements is a set. > > > > For every set, if you removed all elements from the > set, > > you would wind up with a set containing no elements. > > > >> 8. Question: Can the empty set be an element of a > set? > >> Answer: That depends on 2, but the > answer to 2 should make this answer "Yes." > > > > Practically speaking, this depends on your application > -- your > > "universe of consideration". If you have no use > for empty sets > > in your sets, you should not waste any implementation > effort > > on them. > > > > FYI, > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ________________________________________________________ Audi, Fiat, Peugeot, Skoda, Porsche, Toyota, Ford - Kelkoo har brugte biler til en hver smag! Klik her for at sammenligne priser.(http://dk.yahoo.com/r/pat/mmb) ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
