{"title":"N-Way Joint Mutual Exclusion Does Not Imply Any Pairwise Mutual Exclusion for Propositions","authors":"Roy S. Freedman","doi":"arxiv-2409.03784","DOIUrl":null,"url":null,"abstract":"Given a set of N propositions, if any pair is mutual exclusive, then the set\nof all propositions are N-way jointly mutually exclusive. This paper provides a\nnew general counterexample to the converse. We prove that for any set of N\npropositional variables, there exist N propositions such that their N-way\nconjunction is zero, yet all k-way component conjunctions are non-zero. The\nconsequence is that N-way joint mutual exclusion does not imply any pairwise\nmutual exclusion. A similar result is true for sets since propositional\ncalculus and set theory are models for two-element Boolean algebra.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"63 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.03784","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Given a set of N propositions, if any pair is mutual exclusive, then the set
of all propositions are N-way jointly mutually exclusive. This paper provides a
new general counterexample to the converse. We prove that for any set of N
propositional variables, there exist N propositions such that their N-way
conjunction is zero, yet all k-way component conjunctions are non-zero. The
consequence is that N-way joint mutual exclusion does not imply any pairwise
mutual exclusion. A similar result is true for sets since propositional
calculus and set theory are models for two-element Boolean algebra.
给定一个由 N 个命题组成的集合，如果任何一对命题是互斥的，那么所有命题的集合就是 N 路共同互斥的。本文为反义词提供了一个新的一般性反例。我们证明，对于任何由 N 个命题变量组成的集合，都存在 N 个命题，它们的 N 向连词为零，但所有 k 向分量连词都不为零。其结果是，N 向联合互斥并不意味着任何成对互斥。类似的结果也适用于集合，因为命题微积分和集合论是两元素布尔代数的模型。
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