\begin{definition}[Definition:Coarser Subset (Order Theory)]
Let $L = \struct {S, \preceq}$ be a preordered set.
Let $X, Y$ be subsets of $S$.
Then $X$ is '''coarser (subset)''' than $Y$ {{iff}}
:$\forall x \in X: \exists y \in Y: y \preceq x$
\end{definition}