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You are not logged in. #1 20060317 16:00:11
Open and closed caseName a set that is both open and closed. Last edited by Ricky (20060317 16:00:53) "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #2 20060317 18:08:32
Re: Open and closed caseThe set of twin primes. Character is who you are when no one is looking. #3 20060317 18:14:11
Re: Open and closed caseTwas a nice try, but a real solution exists. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #4 20060318 14:02:10
Re: Open and closed caseAnd whats that? Ideas are funny little things, they won't work unless you do. #5 20060318 14:40:52
Re: Open and closed caseAlright, I'll give a hint, and if no one can get it by Sunday, I'll give out the answer. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #6 20060318 15:54:31
Re: Open and closed caseOr a door with a hole. "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #7 20060318 21:58:36
Re: Open and closed caseRicky, if I am thinking right, its the set of complex numbers. Character is who you are when no one is looking. #8 20060319 12:53:45
Re: Open and closed caseI DO love the title... The greatest challenge to any thinker is stating the problem in a way that will allow a solution. Bertrand Russell #9 20060319 15:46:44
Re: Open and closed caseWhat's on second. "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #10 20060319 17:07:23
Re: Open and closed caseI don't know's on third (I think...I can't remember too well) The greatest challenge to any thinker is stating the problem in a way that will allow a solution. Bertrand Russell #11 20060319 18:03:17
Re: Open and closed caseDarthradius got it! "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #12 20060319 23:38:45
Re: Open and closed caseI hardly understand Darthradius! Ideas are funny little things, they won't work unless you do. #13 20060320 03:02:29
Re: Open and closed caseThis requires a bit of knowledge of sets and logic. I'll try to be brief. Also, if you have an if then statement like, "If it is tuesday, then I like cheese", if the first part is false, then no matter what the 2nd part is, the statement is true. Using the same example, if it is Wednesday and I don't like cheese, the statement is still true. If it is Thursday and I do like cheese, the statement is true. The only time it is false if it is Tuesday, and I don't like cheese. The definition for an open set is: A set O is open if for all points a∈O there exists an ∈neighborhood of a. Don't worry about understanding much of that if you don't. The first part is "if for all points a∈O." But there are no points in O (the empty set)! So like the statments above, this statement must be true. Closed set: A set C is closed if the complement of C (everything that isn't in C) is open. The complement of C (the empty set still) is R, the set of all reals. This set is open, and thus, C is closed. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #14 20060320 12:14:52
Re: Open and closed casehooray! The greatest challenge to any thinker is stating the problem in a way that will allow a solution. Bertrand Russell #15 20060320 19:59:44
Re: Open and closed caseWell good time understanding yourself darthradius and thanks to the both of you. I definetly understand better now! Ideas are funny little things, they won't work unless you do. #16 20060321 02:19:54
Re: Open and closed caseThere is one more possible answer. It has something to do with the null set. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #17 20060321 10:31:41
Re: Open and closed casehmmm...I believe the exact same argument would work for the entire set of real numbers, wouldn't it? Last edited by darthradius (20060321 10:33:24) The greatest challenge to any thinker is stating the problem in a way that will allow a solution. Bertrand Russell #18 20060321 10:45:26
Re: Open and closed caseAh, you're right, I forgot the last part:
But what does the boundary points being in the empty set have to do with the set of reals being open? "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #19 20060321 10:53:03
Re: Open and closed caseI was using those theorems... The greatest challenge to any thinker is stating the problem in a way that will allow a solution. Bertrand Russell #20 20060321 10:59:51
Re: Open and closed caseYes, it can be shown that R and the empty set are the only sets which are both open and closed. Took me a little while to work this one out but I think I got it. If you or anyone else wants a hint, let me know. Last edited by Ricky (20060321 11:00:20) "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #21 20060321 12:45:44
Re: Open and closed caseWell, yes...it is fairly obvious The greatest challenge to any thinker is stating the problem in a way that will allow a solution. Bertrand Russell 