Original question and givens
Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.
G001 — Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.
Step-by-step derivation
S001 · implies
Before
raw relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 4], [4, 0]];carrier Fin(6)
After
lexicographically sorted distinct ordered pairs=[[0, 0], [0, 3], [0, 4], [1, 1], [2, 2], [3, 3], [4, 0], [4, 4], [5, 5]]
Conditions: none
List distinct directed pairs lexicographically.Duplicates are invalid;do not silently add reverses,diagonals or a transitive closure.
Step identifiers and dependencies
Rule: complete_relation_no_duplicate_or_implicit_repair · Uses: G001
S002 · implies
Before
first original pair=[0, 0];all composable second endpoints=[0, 3, 4]
After
required transitivity rows [a,b,d,aRd]=[[0, 0, 0, Yes], [0, 0, 3, Yes], [0, 0, 4, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S001
S003 · implies
Before
first original pair=[0, 3];all composable second endpoints=[3]
After
required transitivity rows [a,b,d,aRd]=[[0, 3, 3, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S002
S004 · implies
Before
first original pair=[0, 4];all composable second endpoints=[0, 4]
After
required transitivity rows [a,b,d,aRd]=[[0, 4, 0, Yes], [0, 4, 4, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S003
S005 · implies
Before
first original pair=[1, 1];all composable second endpoints=[1]
After
required transitivity rows [a,b,d,aRd]=[[1, 1, 1, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S004
S006 · implies
Before
first original pair=[2, 2];all composable second endpoints=[2]
After
required transitivity rows [a,b,d,aRd]=[[2, 2, 2, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S005
S007 · implies
Before
first original pair=[3, 3];all composable second endpoints=[3]
After
required transitivity rows [a,b,d,aRd]=[[3, 3, 3, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S006
S008 · implies
Before
first original pair=[4, 0];all composable second endpoints=[0, 3, 4]
After
required transitivity rows [a,b,d,aRd]=[[4, 0, 0, Yes], [4, 0, 3, No], [4, 0, 4, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S007
S009 · implies
Before
first original pair=[4, 4];all composable second endpoints=[0, 4]
After
required transitivity rows [a,b,d,aRd]=[[4, 4, 0, Yes], [4, 4, 4, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S008
S010 · implies
Before
first original pair=[5, 5];all composable second endpoints=[5]
After
required transitivity rows [a,b,d,aRd]=[[5, 5, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S009