A Note on the Union-closed Sets Conjecture

TL;DR

This paper explores a weaker version of the Union-Closed Set Conjecture by examining binary matrices with rows closed under logical operators, focusing on columns with more ones than zeros.

Contribution

It introduces a new approach to the conjecture by analyzing matrices with rows closed under the material conditional, proposing a weaker related conjecture.

Findings

Identifies conditions under which columns have at least as many ones as zeros.

Proposes a weaker conjecture related to the Union-Closed Set Conjecture.

Provides theoretical insights into the structure of such matrices.

Abstract

Let $M$ be a non-zero binary matrix with distinct rows where the rows are closed under certain logical operators. In this article, we investigate the existence of columns containing an equal or greater number of ones than zeros. Specifically, the existence of such columns when the rows of the matrix are closed under $\textit{material conditional}$ leads us to a weaker version of the $\textit{Union-Closed Set Conjecture}$.