Question

Show that the partition formed from congruence classes modulo \(6\) is a refinement of the partition formed from congruence classes modulo \(3\) .

Short answer

Thus, the all set of modulo \(6\) is a subset of some sets of modulo \(3\) .

Step-by-step solution

Given data

The congruence classes modulo \(6\) and congruence classes modulo \(3\) .

Concept used of refinement of partition

If all the set in\(A\)is a subset of some set in\(B\), then\(A\)is known as a refinement of the partition\(B\).

Find the congruence class 

Consider the congruence classes modulo \(6\) a

The six congruence classes of modulo \(6\) are \({(0)_6},{(1)_6},{(2)_6},{(3)_6},{(4)_6}\), and \({(5)_6}\).

Congruence class of \(0\) of modulo \(6\) will be \({(0)_6} = \{ \ldots , - 12, - 6,0,6,12,18, \ldots \} \).

Congruence class of \(1\) of modulo \(6\) will be \({(1)_6} = \{ \ldots , - 11, - 5,1,7,13,19, \ldots \} \).

Congruence class of \(2\) of modulo \(6\) will be \({(2)_6} = \{ \ldots , - 10, - 4,2,8,14,20, \ldots \} \).

Congruence class of \(3\) of modulo \(6\) will be \({(3)_6} = \{ \ldots , - 9, - 3,3,9,15,21, \ldots \} \).

Congruence class of \(4\) of modulo \(6\) will be \({(4)_6} = \{ \ldots , - 8, - 2,4,10,16,22, \ldots \} \).

Congruence class of \(5\) of modulo \(6\) will be \({(5)_6} = \{ \ldots , - 7, - 1,5,11,17,23, \ldots \} \).

These all congruence classes are disjoint. Every integer is in exactly one of them.

Here, let \({A_1}\) be the congruence classes from a partition.

Prove refinement of partition

The three congruence classes of modulo \(3\) are \({(0)_6},{(1)_6}\), and \({(2)_6}\).

Congruence class of \(0\) of modulo \(3\) will be \({(0)_3} = \{ \ldots , - 9, - 6, - 3,0,3,6,9,12, \ldots \} \).

Congruence class of \(1\) of modulo \(3\) will be \({(1)_3} = \{ \ldots , - 8, - 5, - 2,1,4,7,10,13, \ldots \} \).