S001 · implies
Before
complete carrier Fin(6);original relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]
After
set of pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [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, 3, 4, 5]
After
next canonical full class=[0, 3, 4, 5];covered labels=[0, 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, S001
S003 · implies
Before
least uncovered carrier label=1;all labels related from it=[1]
After
next canonical full class=[1];covered labels=[0, 1, 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, S002
S004 · implies
Before
least uncovered carrier label=2;all labels related from it=[2]
After
next canonical full class=[2];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, 3, 4, 5], [1], [2]];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=5;b=4;related=Yes;full classes=[[0, 3, 4, 5], [0, 3, 4, 5]]
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