# 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.

### 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}, {hi}}