A Note on the Union-closed Sets Conjecture

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 $\textit{Union-Closed Set Conjecture}$.