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, 1], [0, 2], [2, 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,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.
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, 1], [0, 2], [2, 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,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.
Step-by-step derivation
S001 · implies
Before
raw relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [2, 0]];carrier Fin(6)
After
lexicographically sorted distinct ordered pairs=[[0, 0], [0, 1], [0, 2], [1, 1], [2, 0], [2, 2], [3, 3], [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
original pair=[0, 0];required reverse=[0, 0]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S001
S003 · implies
Before
original pair=[0, 1];required reverse=[1, 0]
After
reverse present=No
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S002
S004 · implies
Before
original pair=[0, 2];required reverse=[2, 0]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S003
S005 · implies
Before
original pair=[1, 1];required reverse=[1, 1]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S004
S006 · implies
Before
original pair=[2, 0];required reverse=[0, 2]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S005
S007 · implies
Before
original pair=[2, 2];required reverse=[2, 2]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S006
S008 · implies
Before
original pair=[3, 3];required reverse=[3, 3]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S007
S009 · implies
Before
original pair=[4, 4];required reverse=[4, 4]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S008
S010 · implies
Before
original pair=[5, 5];required reverse=[5, 5]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S009