set-theoretical_partition
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| 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 $X$. | 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 $\emptyset$ is $\emptyset$ itself. We speak of the **empty partition**. | + | Note that the only set-theoretical partition of the empty set $\emptyset$ is the empty set $\emptyset$ itself. We speak of the **empty partition**. |
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