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--- set-theoretical_partition [2020/03/12 08:46]
amang created
+++ set-theoretical_partition [2021/11/23 11:56] (current)
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 ====== Set-theoretical partition ======
-Given any set $X$ a **set-theoretical partition** $\pi$ of $X$ is any subset $\pi\subseteq \mathfrak{P}(X)$ of the power set $\mathfrak{P}(X)$ such that $B\cap B'=\emptyset$ for all $B,B'\in \pi$ with $B\neq B'$ and such that $\bigcup \pi=X$. The elements of $\pi$ are called the **blocks** of $\pi$.
+Given any set $X$ a **set-theoretical partition** $\pi$ of $X$ is any subset $\pi\subseteq \mathfrak{P}(X)$ of the power set $\mathfrak{P}(X)$ such that $\emptyset\notin \pi$, such that $B\cap B'=\emptyset$ for all $B,B'\in \pi$ with $B\neq B'$ and such that $\bigcup \pi=X$. The elements of $\pi$ are called the **blocks** of $\pi$.
 Set-theoretical partitions of $X$ are in one-to-one correspondence to //equivalence relations// on $X$. From a partition $\pi$ of $X$ one obtains an equivalence relation $\sim_\pi$ by defining for all $i,j\in X$ that $i\sim_\pi j$ if and only if there exists $B\in \pi$ such that $\{i,j\}\in B$. Conversely, given an equivalence relation $\sim$ on $X$ the set $X/{\sim}=\{ \{j\in X\,\vert\, i\sim j\} \,\vert\, i\in X\}$ of all equivalence classes is a set-theoretical partition of $X$.
+Note that the only set-theoretical partition of the empty set $\emptyset$ is the empty set $\emptyset$ itself. We speak of the **empty partition**.