N-Way Joint Mutual Exclusion Does Not Imply Any Pairwise Mutual Exclusion for Propositions
<|reference_start|>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.<|reference_end|>
@article{freedman2024n-way,
    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},
    archivePrefix={arXiv},
    eprint={2409.03784},
    primaryClass={cs.DM}
}