Quotient Set

The quotient set is the set of equivalence classes formed from a given set under an equivalence relation r.

A Concrete Example

Let’s take a finite set A containing the following elements:

$$ A = \{ 12, 34, 3, 1, 45, 401, 39, 4 \} $$

We define an equivalence relation p on A by grouping together elements that share the same leading digit.

Example. The numbers 3, 34, and 39 all start with the digit 3. The numbers 45, 401, and 4 start with 4. The numbers 12 and 1 start with 1.

Based on this relation, we can form three distinct equivalence classes in A:

$$ [12] = \{ 12, 1 \} $$

$$ [34] = \{ 34, 3, 39 \} $$

$$ [45] = \{ 45, 401, 4 \} $$

The quotient set is the set of all these equivalence classes:

$$ A/p = \{ [12], [34], [45] \} $$

This is read as “A mod p” or “A modulo p.”