From sbelknap@UIC.EDU Tue Mar 07 09:14:42 2000 Received: (qmail 15139 invoked from network); 7 Mar 2000 17:14:57 -0000 Received: from unknown (10.1.10.26) by m1.onelist.org with QMQP; 7 Mar 2000 17:14:57 -0000 Received: from unknown (HELO eeyore.cc.uic.edu) (128.248.171.51) by mta1.onelist.com with SMTP; 7 Mar 2000 17:14:57 -0000 Received: from [128.248.250.241] (mac0.uicomp.uic.edu [128.248.250.241]) by eeyore.cc.uic.edu (8.9.3/8.9.3) with ESMTP id LAA14023 for ; Tue, 7 Mar 2000 11:12:02 -0600 (CST) Mime-Version: 1.0 X-Sender: sbelknap@mailserv.uic.edu Message-Id: Date: Tue, 7 Mar 2000 11:14:35 -0600 To: lojban@onelist.com Subject: Non-circular definition of final club with pc's addendums included Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-eGroups-From: Steven Belknap From: Steven Belknap X-Yahoo-Message-Num: 2228 A student at Yale may belong to zero or more clubs. Some clubs are final clubs. A final club is defined as "a club such that membership in it precludes membership in any other final club". 0. There are at least two distinct clubs. (If there were not, there would not be a solution, and we are given that there is a solution.) 1. Call a set, s, of clubs preclusive if it contains more than one member and if being a member of any one of the clubs in s precludes being a member of any other club in s. 2. Call a set, m, of clubs maximally preclusive if it is preclusive and every proper superset of m is not preclusive. 3. There is one and only one nonempty set, m. (We are told that some clubs are final clubs. If there were no nonempty sets m, that would violate the conditions of the problem. If there were more than one set m, than it would not be possible to know which set m contained the final clubs, and a definition of final club would be impossible. Since the problem contains a definition, albeit a circular one, a definition must be possible and there must be only one nonempty set, m.) 4. Call a club, c, a final club iff it is a member of m. QED There are some singularities to consider. For example, what if no Yale student is a member of any club? Can there be a final club? I think that there can not be, but it is not clear. I interpret the question as requiring that at least two Yale students are members of clubs, and that each student would then be a member of a final club. But the question is vague about how to consider clubs which have rules and no members. Also, pc points out that the question does not exclude the possibility that some clubs are not distinct from other clubs. Also, do we base our understanding of what preclusive means on what the rules of a club are or what the behavior of a club is? co'o mi'e stivn Steven Belknap, M.D. Assistant Professor of Clinical Pharmacology and Medicine University of Illinois College of Medicine at Peoria