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.
Example
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}, {h, i}}