An arbitrary partition $F$, for some equivalence relation $R$, importantly does not contain any set $X$ such that $X = \emptyset$. However, is it possible for a constituent equivalence class, $[x]_R$ to contain the null set? I don't mean this in the sense that $[x]_R = \emptyset$, but whether $\exists [x]_R : \emptyset \subset [x]_R$. If so, will $[x]_R$ be contained in a partition?
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$\emptyset\subseteq A$ is true for all sets $A$, including the ones that arise as classes of some equivalence (which would technically be all non-empty sets). – Sassatelli Giulio Jan 31 '24 at 07:17
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Do you ask whether a partitioning can contain the empty set? This is a matter of definition, we can certainly make such definition. But this will only make our lifes harder, because it doesn't bring anything useful, and will require "is nonempty" check everywhere. – freakish Jan 31 '24 at 08:30