1. What a Partition of a Set Means
A partition of a set is a way of breaking the set into separate, non-overlapping groups. Each element of the set must belong to exactly one of these groups. No element is left out, and no element appears in two groups at once.
These groups are often called blocks of the partition.
2. Basic Requirements for a Partition
A collection of subsets forms a partition of a set A if these conditions hold:
2.1. Condition 1 — Each Subset Is Non-Empty
No block of the partition can be empty.
2.2. Condition 2 — Subsets Cover the Whole Set
Every element of A must appear in at least one block.
2.3. Condition 3 — Subsets Do Not Overlap
Any two different blocks must be disjoint — they cannot share elements.
3. Example of a Partition
Let A = {1,2,3,4,5,6}. A possible partition of A is:
\{\{1,2\},\{3,4\},\{5,6\}\}
Each element appears exactly once, the subsets do not overlap, and none of them is empty.
4. How Equivalence Relations Create Partitions
An equivalence relation naturally breaks a set into disjoint groups. Each element belongs to an equivalence class, and these equivalence classes together form a partition of the set.
4.1. Connection Between the Two Ideas
If R is an equivalence relation on A, then:
- Each element a forms a class [a],
- All classes together cover A,
- No two different classes overlap.
4.2. Key Point
Equivalence relations and partitions describe the same structure in different ways.
5. Illustration Using Equivalence Classes
Suppose A contains shapes, and the relation is “has the same colour as”. Then:
- All red shapes form one class,
- All blue shapes form another class,
- All green shapes form another class.
These classes do not overlap and together form a partition of the set of shapes.
6. Example: Partition from a Modulo Relation
On the integers, define a relation:
a \sim b \; \text{if} \; a \equiv b \pmod{3}
This relation splits the integers into three equivalence classes:
- [0] → numbers with remainder 0 mod 3
- [1] → numbers with remainder 1 mod 3
- [2] → numbers with remainder 2 mod 3
Together, these three classes form a partition of all integers.
7. Why Partitions Are Useful
Partitions help organise sets into meaningful groups based on shared properties. Equivalence relations give a precise rule for forming these groups. This idea is widely used in algebra, number theory, geometry, and classification problems.