Partitions

In mathematics, a partition of a set means dividing the set's elements into smaller, non-empty parts such that every element goes into exactly one part, and together these parts cover the entire set. In other words: the partition 'divides' the whole set into disjoint groups.

The elements of the partition are called 'subsets' or 'blocks'. Every element belongs to exactly one block, cannot be in two at once, and no element is left out.

Formal Definition

Let A be a set. A P is a partition of A if P is a collection of sets for which the following hold:

• Every part is non-empty: ∀ S ∈ P, S ≠ ∅.
• The parts are disjoint: ∀ S, T ∈ P, if S ≠ T, then S ∩ T = ∅.
• The union of the parts is A: ∪_{S ∈ P} S = A.

Everyday Examples

• Dividing students into classes: each class is a block, students cannot be in two classes, and all students are assigned.
• Sorting fruits into baskets by type: apples in one, oranges in another – no overlap, full coverage.
• Dividing a cake into slices: each slice is a part, no piece left out, no overlapping slices.

Mathematical Examples

• Partition of {1,2,3,4} into {{1,2}, {3,4}}: two blocks, disjoint, cover the whole set.
• Singleton partition: {{1}, {2}, {3}, {4}} – each element in its own block.
• Trivial partition: {{1,2,3,4}} – the whole set as one block.

Connection to Equivalence Relations

There is a deep connection between equivalence relations and partitions. Every equivalence relation induces a partition: the equivalence classes are the blocks of the partition, where two elements are in the same subset if they are equivalent.

Conversely, every partition defines an equivalence relation: two elements are equivalent if they belong to the same subset.

Therefore, equivalence relations and partitions represent two sides of the same mathematical phenomenon: one in the language of connections, the other in the language of set division, expressing equivalence.

Summary

A partition is a division of a set's elements where there is no overlap and no missing element. It is closely related to equivalence relations: every equivalence relation determines a partition, and every partition corresponds to an equivalence relation.

Practice Exercise