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Formal Definition
A partition of a set $S$ is a collection of nonempty subsets of $S$ such that every element of $S$ is in exactly one of the subsets. The subsets are the cells of the partition.
Informal Definition
A partition of a set $S$ is the collection of nonempty subsets of $S$ that contain all elements of the original set exactly once.
Example(s)
Splitting $\mathbb{Z}^+$ into the subset of even positive integers (those divisible by 2) and the subset of odd positive integers (those leaving a remainder of 1 when divided by 2), we obtain a partition of $\mathbb{Z}^+$ into two cells.
Non-example(s)
Splitting the set $S=\{2, 6, 9\}$ into the subset of integers divisible by 2 and integers divisible by 3 yields: $\{2,6\}, \{6,9\}$. The element $6$ appears twice, therefore this is not a partition of $S$.
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