Partition of a Set
Set [latex]\mathcal{P}[/latex] of disjoint subsets of a set E with these two properties:
- Each subset of [latex]\mathcal{P}[/latex] is not empty;
- The union of all the subsets of E in [latex]\mathcal{P}[/latex] 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 [latex]\mathcal{R}[/latex] defined in a set E causes in this set a partition [latex]\wp[/latex] into classes of equivalences. The set [latex]\wp[/latex] 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 [latex]\wp[/latex] of a set E:
- E = {a, b, c, d, e, f, g h, i}
- [latex]\wp[/latex] = {{a, b, c, d}, {e, f}, {g}, {h, i}}