Partition

Last updated Nov 1, 2022

Let $X$ be a Set. A Partition $\mathcal{S} \subset \mathcal{P}(X)$ of $X$ is a collection of Mutually Disjoint Nonempty subsets of $X$ that cover $X$. That is:

1. $\emptyset \not\in \mathcal{S}$.
2. $\forall A, B \in \mathcal{S}$, either $A \cap B = \emptyset$ or $A = B$.
3. $\bigsqcup\limits_{A \in \mathcal{S}} A = X$.

• For condition (2), observe that the two possibilities are Mutually Exclusive. Since $\emptyset \not\in \mathcal{S}$, if $A, B \in \mathcal{S}$ and $A = B$, then $\exists x \in A \cap B$.
• Condition (2) can equivalently be written as $\forall A, B \in \mathcal{S} (A \cap B \neq \emptyset \Rightarrow A = B)$. This follows from the definition of Logical Implication.

• Power Set
• Empty Set