Original question and givens
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=3, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
G001 — Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=3, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
Step-by-step derivation
S001 · implies
Before
complete carrier Fin(6);raw relation=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]
After
sorted distinct ordered relation=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]];missing reflexive=[];missing reverses=[];missing transitive triples=[]
Conditions: none
Qualify reflexivity,symmetry and every composable pair's transitivity before forming classes.Do not silently deduplicate invalid inputs or infer equivalence from samples or connected clustering.
Step identifiers and dependencies
Rule: equivalence_relation_complete_parent_qualification · Uses: G001
S002 · implies
Before
first class:representative=3;carrier candidate=0
After
ordered relation pair present=No;class membership=No
Conditions: none
By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.
Step identifiers and dependencies
Rule: equivalence_class_every_carrier_element · Uses: G001, S001
S003 · implies
Before
first class:representative=3;carrier candidate=1
After
ordered relation pair present=No;class membership=No
Conditions: none
By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.
Step identifiers and dependencies
Rule: equivalence_class_every_carrier_element · Uses: G001, S002
S004 · implies
Before
first class:representative=3;carrier candidate=2
After
ordered relation pair present=Yes;class membership=Yes
Conditions: none
By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.
Step identifiers and dependencies
Rule: equivalence_class_every_carrier_element · Uses: G001, S003
S005 · implies
Before
first class:representative=3;carrier candidate=3
After
ordered relation pair present=Yes;class membership=Yes
Conditions: none
By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.
Step identifiers and dependencies
Rule: equivalence_class_every_carrier_element · Uses: G001, S004
S006 · implies
Before
first class:representative=3;carrier candidate=4
After
ordered relation pair present=Yes;class membership=Yes
Conditions: none
By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.
Step identifiers and dependencies
Rule: equivalence_class_every_carrier_element · Uses: G001, S005
S007 · implies
Before
first class:representative=3;carrier candidate=5
After
ordered relation pair present=Yes;class membership=Yes
Conditions: none
By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.
Step identifiers and dependencies
Rule: equivalence_class_every_carrier_element · Uses: G001, S006
S008 · implies
Before
first class:all carrier labels tested with representative=3
After
complete class in increasing order=[2, 3, 4, 5]
Conditions: none
The entire carrier has been inspected;this list is the representative's full equivalence class,not a local sample.
Step identifiers and dependencies
Rule: class_complete_inventory_not_sample · Uses: G001, S007
S009 · implies
Before
second class:representative=4;carrier candidate=0
After
ordered relation pair present=No;class membership=No
Conditions: none
By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.
Step identifiers and dependencies
Rule: equivalence_class_every_carrier_element · Uses: G001, S008
S010 · implies
Before
second class:representative=4;carrier candidate=1
After
ordered relation pair present=No;class membership=No
Conditions: none
By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.
Step identifiers and dependencies
Rule: equivalence_class_every_carrier_element · Uses: G001, S009
S011 · implies
Before
second class:representative=4;carrier candidate=2
After
ordered relation pair present=Yes;class membership=Yes
Conditions: none
By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.
Step identifiers and dependencies
Rule: equivalence_class_every_carrier_element · Uses: G001, S010
S012 · implies
Before
second class:representative=4;carrier candidate=3
After
ordered relation pair present=Yes;class membership=Yes
Conditions: none
By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.
Step identifiers and dependencies
Rule: equivalence_class_every_carrier_element · Uses: G001, S011
S013 · implies
Before
second class:representative=4;carrier candidate=4
After
ordered relation pair present=Yes;class membership=Yes
Conditions: none
By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.
Step identifiers and dependencies
Rule: equivalence_class_every_carrier_element · Uses: G001, S012
S014 · implies
Before
second class:representative=4;carrier candidate=5
After
ordered relation pair present=Yes;class membership=Yes
Conditions: none
By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.
Step identifiers and dependencies
Rule: equivalence_class_every_carrier_element · Uses: G001, S013
S015 · implies
Before
second class:all carrier labels tested with representative=4
After
complete class in increasing order=[2, 3, 4, 5]
Conditions: none
The entire carrier has been inspected;this list is the representative's full equivalence class,not a local sample.
Step identifiers and dependencies
Rule: class_complete_inventory_not_sample · Uses: G001, S014
S016 · implies
Before
class a=[2, 3, 4, 5];class b=[2, 3, 4, 5];intersection=[2, 3, 4, 5]
After
equal-or-disjoint=Yes;classification=equal
Conditions: none
A common x with aRx and bRx gives aRb by symmetry/transitivity,then representative invariance makes the classes equal.Otherwise the intersection is empty.This uses qualified equivalence axioms and constructs no third class type or quotient operation.
Step identifiers and dependencies
Rule: classes_equal_or_disjoint_full_axiom_reason · Uses: G001, S015