Original question and givens
Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [2, 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. a=4, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).
G001 — Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [2, 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. a=4, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).
Step-by-step derivation
S001 · implies
Before
complete carrier Fin(6);original relation pairs=[[0, 0], [1, 1], [1, 2], [2, 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], [1, 1], [1, 2], [2, 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]
After
next canonical full class=[0];covered labels=[0]
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=1;all labels related from it=[1, 2]
After
next canonical full class=[1, 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 representatives a=4;b=1;related=No;full classes=[[3, 4, 5], [1, 2]]
After
related implies same class=Yes;no_quotient_operation_claim
Conditions: none
If a~b then their classes agree.When unrelated,the original implication is true without asserting class equality.This is representative independence of the class,not well-definedness of an unspecified quotient operation.
Step identifiers and dependencies
Rule: quotient_representative_implication_no_algebra_operation · Uses: G001, S005
Result
(4, 1, No, [3, 4, 5], [1, 2], Yes, no_quotient_operation_claim)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.