See http://en.wikipedia.org/wiki/Multiset for 'sets' having the same element more than once.
--- Den man 3/8/09 skrev Kip Murray <[email protected]>: > Fra: Kip Murray <[email protected]> > Emne: Re: [Jprogramming] J Sets > Til: "Programming forum" <[email protected]> > Dato: mandag 3. august 2009 07.16 > Bill, taking your points in order: > > Question 1. In abstract set theory, "set" and "is an > element of" are undefined, > but an implementor of a concrete model for set theory must > define which objects > he is using to represent sets. That is, an answer for > Question 1 is required of > an implementor of a concrete model. > > Question 2. You are correct, it is the relation "is > an element of" that must be > defined. I misstated Question 2. > > Question 7. That's exactly right. > > Multiple occurrences of an element. I had a teacher > who said one day, "You are > not going to like this, but we are going to allow a number > to be a member more > than once, and we will take into account 'multiple > membership' in finding the > sum of the numbers in a finite set. We will then use > the idea of sum for a > finite set to define what is meant by a sum for the numbers > in an infinite set. > Not every infinite set of numbers has a sum." > So, multiple occurrences of the > same element are sometimes allowed, but are frowned > on! Outside of that one > course, they were never allowed in my training, and I > learned later it is more > acceptable to talk about a sum of an infinite sequence (in > sequences repetitions > _are_ allowed), and that the teacher was using "sum for a > set" to finesse > absolute convergence. ...more than you wanted to > know In brief, I think of the > names you wrote as different names for the same set of only > three numbers. > > By the way, what does iirc mean? "if I recall"? > > Kip > > > bill lam wrote: > > On Sun, 02 Aug 2009, Kip Murray wrote: > >> Tracy, I abandon my model of finite set theory to > absorb the attractive proposal > >> of Dan and Fraser. Fraser's set-creating > verb is > >> > >> Set =: (/:~)@~. NB. sorted nub > >> > >> and I take your suggestion to be, maybe "sorted" > can be dispensed with. I have > >> to punt to the group while I absorb. > >> > >> I propose that any model of set theory should > complete the following "catechism": > >> > >> 1. Question: What is a set? > >> Answer: > > > > iirc this should be undefined. > > > >> 2. Question: What is an element of a set? > >> Answer: > > > > iirc this should be undefined, but you can/should > define a test for > > membership, eg. `e.' in J > > > >> 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. > >> > >> 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. > >> > >> 5. Question: What is an empty set? > >> Answer: The empty set, > called phi, is the set which has no element. > >> > >> 6. Question: Why did you say "The" empty set? > >> Answer: If H is a set > which has no element, then there is no element of H > >> > that is not an element of phi, and there > is no element of phi that is > >> > not an element of H -- because there is no > element of H or phi. By 3, > >> > H is phi. Thus phi is the only empty > set. > >> > >> (It is standard to interpret "Each one is" to mean > "Not one is not".) > >> > >> 7. Question: Why is the empty set a subset of > every set? > >> Answer: > > > > using negation of Q4: no element of empty set which is > not an element of > > any set. Or there is no counter example that there > exits one element of > > phi that not belonged to other sets. > > > > Last night I googled and there is quotation that > elements inside a set > > need not be distinct, but 2 sets are consider equal if > they only > > differ in repeated elements eg. > > > > { 1 2 3 } = { 1 1 2 3} = { 1 1 2 2 3 3 3} > > > > the equality follows from Q3. > > > > -- > > regards, > > ==================================================== > > GPG key 1024D/4434BAB3 2008-08-24 > > gpg --keyserver subkeys.pgp.net --recv-keys 4434BAB3 > > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > __________________________________________________________ Find din nye laptop på kelkoo.dk. Se de gode tilbud her - http://dk.yahoo.com/r/pat/mm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
