ThmDex – An index of mathematical definitions, results, and conjectures.
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Zermelo-Fraenkel set theory
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Set partition
Definition D83
Proper set partition
Formulation 0
Let $X$ be a D11: Set.
A D11: Set $\mathcal{S} \subseteq \mathcal{P}(X)$ is a proper set partition of $X$ if and only if
(1) \begin{equation} X = \cup \mathcal{S} \end{equation} D4984: Tight set cover
(2) \begin{equation} \forall \, E, F \in \mathcal{S} \left( E \neq F \quad \implies \quad E \cap F = \emptyset \right) \end{equation} D1681: Disjoint set collection
(3) \begin{equation} \forall \, E \in \mathcal{S} : E \neq \emptyset \end{equation}
Formulation 1
Let $X$ be a D11: Set.
A D5143: Set partition $\mathcal{S} \subseteq \mathcal{P}(X)$ of $X$ is a proper set partition of $X$ if and only if \begin{equation} \forall \, E \in \mathcal{S} : E \neq \emptyset \end{equation}
Children
D447: Open set partition
D330: Proper set partition relation
D2910: Set of proper set partitions