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You are not logged in. #26 20070224 12:32:17
Re: Set TheoryAnd there in lies the problem. You are operating under what is called (I swear, this is the actual term, I'm not trying to be insulting), "naive set theory". It is the concept that any collection can be a set. Naive set theory is called so because it is, well, naive. It leads to contradictions and paradoxes. "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..." #27 20070224 12:45:39
Re: Set TheoryI started before you posted, and that isn't without loss of generality, because we assume U to be the set of all sets, so it makes no difference whether the set exists, because it won't be included in here. As for ZFC, yadda yadda blah blah Trussian. It still holds because it's not predicate dependent, union can still be expressed as 2ary. The axiom of union is present for essentially that purpose, to axiomate that such a set is possible to exist. The union operator however is only named because it's serves the purpose that the union axiom expresses is possible. Last edited by Sekky (20070224 12:47:53) #28 20070224 15:45:14
Re: Set TheoryGuys, this is such a great discussion, but it is in the Formulas section, so due for deletion at some point! "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #29 20070224 17:32:13
Re: Set TheorySekky, we're starting to beat around the bush here. Please come up with a binary operator definition for the union of two (possibly different) sets. "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..." #30 20070227 01:12:04
Re: Set Theory
Hello? The definition is right there in my last post! Defined as binary Last edited by Sekky (20070227 01:12:22) #31 20070227 03:05:58
Re: Set TheoryI'm sorry, but that is not a binary operation. A binary operation is a mapping from AxA to A. "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..." #32 20070227 06:01:21
Re: Set Theory
Hello!? A binary operation is an operation whose arity is two This doesn't get much clearer, if this still doesn't make sense, I suggest you study group theory before even considering trying to reiterate the exact point I'm making as an attempt to argue back, because you're not demonstrating much ability by repeatedly saying "A CROSS A MAPS TO A LOLZ", because I've already proved that it can. Simply because the elements are sets doesn't make it any different to any other binary operation you would care to mention. Last edited by Sekky (20070227 06:05:06) #33 20070227 07:41:35
Re: Set TheoryYou have still yet to define union in terms of a binary operation. Please, answer the question. You state that union is a binary operation, all I ask is that you define it as such. I'm not looking for examples, I'm look for you to define union as a mapping from a set AxA to A. "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..." #34 20070227 08:01:40
Re: Set Theorydefining union from AxA to A? *edit*: this is wrong! Last edited by kylekatarn (20070227 10:29:29) #35 20070227 10:05:27
Re: Set TheoryIn your definition, how is A defined? "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..." #36 20070227 10:29:01
Re: Set TheoryI see your point, the definition doesn't seem correct after all. #37 20070227 10:57:02
Re: Set TheoryOddly enough, I didn't think I had a point. I was really just trying to understand your definition. But perhaps it can be saved. This is defined over all of AxA in the same way we can take the union of any two sets. It is also uniquely defined, as all unions are. Only one problem. The universal set contains all sets. So the universal set must contain itself, as it is a set. But this is not allowed by the Axiom of Regularity* (under ZFC set theory). So such a universal set can't exist. *Axiom of Regularity states: This is saying in a general way that A can not be in A. Proof by contradiction. Assume A is in A. Then A is in {A} intersect A. There must exist a b in A such that b intersect {A} = null. Since the only element of {A} is A itself, it must be that b = A. So we replace "b intersect {A} = null" with "A intersect {A} = null". So A is in {A} intersect A and A intersect {A} is null. Contradiction. "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..." #38 20070227 11:19:33
Re: Set TheoryRemove this after cleanup: Last edited by John E. Franklin (20070227 12:22:03) igloo myrtilles fourmis #39 20070228 05:54:07
Re: Set Theory
Are you serious? You want me to list every possible tuple in existance? You're clinically insane, I'd like to see you do that for an infinite set. There you go, have fun writing it out for every possible set imaginable. My previous posts are the method in which it is defined, not by the mapping. Are you telling me scalar addition is defined by writing out every single possible tuple and a permutation? Of course not, it's defined by a couple of predicate rules. #40 20070228 07:33:24
Re: Set TheoryYet again, you post an example. What I am asking for is a general definition. For example, a general definition for addition on the natural numbers is: Where S(n) is the successor function of n. This defines addition for all natural numbers. I am looking for the same general definition for union. There are two ways I see that you can attempt to do so: 1. Take the universal set and create a mapping. Problem is, the universal set doesn't exist under ZFC. 2. Take a set which contains the elements of sets A and B, and create a mapping. Problem is, you can't define a set, in general, which contains the elements of A and B without taking the union of them. "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..." #41 20070228 07:43:19
Re: Set Theory
No, you don't take a set which contains the elements of sets A and sets B. You take the power set of any given set and sets A and B will be elements of that set as a result! If sets A and B don't exist, it's because the elements that constitute them didn't exist in the given set. Notice how the set is given? Because it's a given set. I can't stress this enough. You don't need to define for general A and B because A and B are derived from whatever set you decide to power in the first place. I love how you can easily assume a very composite predicate such as the successor function definition of the natural numbers, but not a simple axiom such as the power set. It exists for a reason, and this is it. #42 20070228 07:54:22
Re: Set Theory
Ok, define addition for the set {{},{1},{2},{1,2}} #43 20080110 07:41:14
Re: Set Theoryhi ganesh, first of all i would like to thank you for this great stuff. Any guy will find it very useful in any mode of his life. i fell lucky to find this site. Thanks once again and keep going........ #44 20080110 18:44:06
Re: Set TheoryThanks, medivijaysagar! Character is who you are when no one is looking. #45 20081125 17:12:37
Re: Set TheoryCartesian Products 2) Character is who you are when no one is looking. 