Partition of a Set

Partition of a Set

Set \(\mathcal{P}\) of disjoint subsets of a set E with these two properties:

  • Each subset of \(\mathcal{P}\) is not empty;
  • The union of all the subsets of E in \(\mathcal{P}\) is equal to E.

  • A partition of a set is a kind of classification of the elements in a set by an equivalence relationship.
  • The concepts of partitions, equivalence relations, and quotient sets are closely linked. In fact, any equivalence relation \(\mathcal{R}\) defined in a set E causes in this set a partition \(\wp\) into classes of equivalences. The set \(\wp\) of these classes is called the quotient set of E by the equivalence relation ℜ and is noted as: E / ℜ.


Consider the partition \(\wp\) of a set E:

  • E = {a, b, c, d, e, f, g h, i}
  • \(\wp\) = {{a, b, c, d}, {e, f}, {g}, {hi}}

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