From jjllambias@hotmail.com Sun Mar 05 12:42:16 2000 Received: (qmail 12227 invoked from network); 5 Mar 2000 20:42:31 -0000 Received: from unknown (10.1.10.26) by m3.onelist.org with QMQP; 5 Mar 2000 20:42:31 -0000 Received: from unknown (HELO hotmail.com) (216.33.241.198) by mta1.onelist.com with SMTP; 5 Mar 2000 20:42:31 -0000 Received: (qmail 49052 invoked by uid 0); 5 Mar 2000 20:42:31 -0000 Message-ID: <20000305204231.49051.qmail@hotmail.com> Received: from 200.41.247.43 by www.hotmail.com with HTTP; Sun, 05 Mar 2000 12:42:31 PST X-Originating-IP: [200.41.247.43] To: lojban@onelist.com Subject: Re: [lojban] Final clubs finally in Lojban Date: Sun, 05 Mar 2000 12:42:31 PST Mime-Version: 1.0 Content-Type: text/plain; format=flowed X-eGroups-From: "Jorge Llambias" From: "Jorge Llambias" la stivn cusku di'e >1. Call a set, s, of clubs preclusive 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 of m is not preclusive. Nice! More compact than my definitions. Using {girzu} for "club", {klesi} for "set", {vasru selkle} for "proper superset", and an interesting interpretation of {natfe} for "preclude": i pamai ro da poi klesi le'i girzu zo'u ca'e da natfe klesi ijo ro de poi cmima da ku'o ro di poi cmima da gi'enai du de zo'u le ka ce'u cmima de cu natfe le ka ce'u cmima di i remai ro da poi klesi le'i girzu zo'u ca'e da nafrai klesi ijo da natfe klesi ijebo ro de poi vasru selkle da na natfe klesi >3. Call a set, f, of clubs the set of final clubs if f is the largest >set of maximally preclusive clubs. But f can't always be determined by that. In some situations there is no single largest maximally preclusive set. For example, we could have the sets {A,B}, {B,C} and {D,E} all being maximally preclusive. (And I insist that this "largest" condition was not a premise of the problem as posed.) >4. Call a club, c, a final club iff it is a member of f. Yes, I forgot that last step. Now that we have the definitions, let me prove a theorem: Theorem: Every club belongs to at least one maximally preclusive set. Proof: By construction: start with the club in question, and examine every other club in any order. If membership in the club under examination precludes membership in the initial club and in every other already accepted club, then accept it, else reject it. The set of accepted clubs plus the initial club is by construction a maximally preclusive set. QED. Corollary: If only one maximally preclusive set exists, then it must be the set of all clubs. Proof: Since every club belongs to at least one m.p.s., then when there is only one m.p.s. all clubs must belong to it. co'o mi'e xorxes ______________________________________________________ Get Your Private, Free Email at http://www.hotmail.com