Original question and givens
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [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=0, b=1. Give (a,b,related?,class_a,class_b,invariance?).
G001 — Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [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=0, b=1. Give (a,b,related?,class_a,class_b,invariance?).
Step-by-step derivation
S001 · implies
Before
complete carrier Fin(6);raw relation=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [5, 4], [5, 5]]
After
sorted distinct ordered relation=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [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=0;carrier candidate=0
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, S001
S003 · implies
Before
first class:representative=0;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=0;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=0;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=0;carrier candidate=4
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, S005
S007 · implies
Before
first class:representative=0;carrier candidate=5
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, S006
S008 · implies
Before
first class:all carrier labels tested with representative=0
After
complete class in increasing order=[0, 2, 3]
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=1;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=1;carrier candidate=1
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, S009
S011 · implies
Before
second class:representative=1;carrier candidate=2
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, S010
S012 · implies
Before
second class:representative=1;carrier candidate=3
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, S011
S013 · implies
Before
second class:representative=1;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=1;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=1
After
complete class in increasing order=[1, 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
a=0;b=1;aRb=No;class a=[0, 2, 3];class b=[1, 4, 5]
After
if aRb then classes equal=Yes;not an assertion that aRb holds
Conditions: none
If aRb,symmetry gives bRa and transitivity sends aRx to bRx;the reverse inclusion follows similarly.If a,b are unrelated,the requested implication is vacuously true and does not assert equal classes.
Step identifiers and dependencies
Rule: representative_invariance_conditional_not_equal_premise · Uses: G001, S015