Partition

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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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