Partition

Definition

A partition splits a set into groups.

The groups must satisfy three conditions:

• no group is empty
• no element appears in more than one group
• every element from the original set appears in some group

Partition = complete non-overlapping grouping.

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Simple Example

Suppose:

A = {1, 2, 3, 4, 5, 6}

A partition of A is:

{{1, 3, 5}, {2, 4, 6}}

This groups the set into:

• odd numbers
• even numbers

Every element appears once.

No element appears in both groups.

A partition organizes a set without duplication or omission.

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Non-Example: Overlap

This is not a partition:

{{1, 2}, {2, 3}, {4}}

Why?

Because 2 appears in two groups.

A partition cannot overlap.

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Non-Example: Missing Element

This is not a partition of:

A = {1, 2, 3, 4}

{{1, 2}, {3}}

Why?

Because 4 is missing.

A partition must cover the whole set.

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Partition And Equivalence

Equivalence relations naturally create partitions.

Example:

a relates to b if a and b have the same remainder when divided by 2

This creates:

{1, 3, 5}
{2, 4, 6}

Each equivalence class is one group in the partition.

Equivalence relations group elements; partitions are the resulting groups.

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Final Summary

• A partition splits a set into groups.
• Each group must be non-empty.
• Groups must not overlap.
• The groups must cover the whole original set.
• Equivalence relations create partitions.
• Partitions make grouping precise.