Original question and givens
Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=1. Give (element,class).
G001 — Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=1. Give (element,class).
Step-by-step derivation
S001 · implies
Before
complete carrier Fin(6);original relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]
After
set of pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]];reflexive/symmetric/transitive=[Yes, Yes, Yes]
Conditions: none
Read pairs with set semantics;duplicate entries do not change the relation.Check reflexivity,symmetry and transitivity before forming the quotient.Connected components of a non-equivalence relation are not its quotient classes.
Step identifiers and dependencies
Rule: quotient_complete_equivalence_not_connectedness_only · Uses: G001
S002 · implies
Before
least uncovered carrier label=0;all labels related from it=[0, 1]
After
next canonical full class=[0, 1];covered labels=[0, 1]
Conditions: none
List every element in the class in carrier order,and order classes by the least not-yet-covered element.A quotient element is the whole class,not merely its representative label.
Step identifiers and dependencies
Rule: quotient_full_classes_in_least_representative_order · Uses: G001, S001
S003 · implies
Before
least uncovered carrier label=2;all labels related from it=[2]
After
next canonical full class=[2];covered labels=[0, 1, 2]
Conditions: none
List every element in the class in carrier order,and order classes by the least not-yet-covered element.A quotient element is the whole class,not merely its representative label.
Step identifiers and dependencies
Rule: quotient_full_classes_in_least_representative_order · Uses: G001, S002
S004 · implies
Before
least uncovered carrier label=3;all labels related from it=[3, 4, 5]
After
next canonical full class=[3, 4, 5];covered labels=[0, 1, 2, 3, 4, 5]
Conditions: none
List every element in the class in carrier order,and order classes by the least not-yet-covered element.A quotient element is the whole class,not merely its representative label.
Step identifiers and dependencies
Rule: quotient_full_classes_in_least_representative_order · Uses: G001, S003
S005 · implies
Before
all complete canonical classes=[[0, 1], [2], [3, 4, 5]];original carrier=[0, 1, 2, 3, 4, 5]
After
cover whole carrier=Yes;pairwise disjoint=Yes;class count=3
Conditions: none
Equivalence classes cover the original carrier and are pairwise disjoint,so each element lies in exactly one class.The empty family on an empty carrier also satisfies empty coverage and disjointness;class counts alone are not the partition test.
Step identifiers and dependencies
Rule: quotient_partition_cover_and_disjoint_not_count_only · Uses: G001, S004
S006 · implies
Before
original element=1;canonical quotient elements=[[0, 1], [2], [3, 4, 5]]
After
projection sends1 to whole class=[0, 1]
Conditions: none
The natural projection sends an element to its entire equivalence class,not its least element.No quotient addition or multiplication has been defined.
Step identifiers and dependencies
Rule: quotient_projection_element_to_full_class · Uses: G001, S005