Definition A partition $\Pi$ of a set $A$ is a set of nonempty subsets of $A$ that is disjoint and exhaustive, i.e.,
(a) no two different sets in $\Pi$ have any common elements, and
(b) each element of $A$ is in some set in $\Pi$
This definition doesn't explicitly say $A$ is nonempty. And to me this is interpreted in two ways.
- "nonempty subsets of $A$" implies $A$ is nonempty.
- When $A$ is the empty set some kind of vacuous true occurs(I can't make a precise argument but it seems). So empty set has a partition(but if so, what is a partition of $\emptyset$?).
Best Answer
Yes, it's true that the emptyset as no non-empty subsets. Curiously this does not imply that there is no partition of $\emptyset$.
Note that every element of $\emptyset$ is a non-empty subset of $\emptyset$, vacuously. The (non-existent) elements are pairwise disjoint and have union equal to $\emptyset$, qed.