N 向联合互斥并不意味着命题的任何成对互斥

Roy S. Freedman
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引用次数: 0

摘要

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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N-Way Joint Mutual Exclusion Does Not Imply Any Pairwise Mutual Exclusion for Propositions
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.
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