N-Way Joint Mutual Exclusion Does Not Imply Any Pairwise Mutual Exclusion for Propositions
Discrete Mathematics
2024-09-09 v1
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.
Cite
@article{arxiv.2409.03784,
title = {N-Way Joint Mutual Exclusion Does Not Imply Any Pairwise Mutual Exclusion for Propositions},
author = {Roy S. Freedman},
journal= {arXiv preprint arXiv:2409.03784},
year = {2024}
}
Comments
10 pages, 4 figures