Question 12327: Every Carrier Element Needs Its Diagonal Pair

Original question and givens

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2]]. 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 missing diagonal element labels in increasing order,reflexive?).

• G001 — Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2]]. 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 missing diagonal element labels in increasing order,reflexive?).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-equivalence-relation

Step-by-step derivation

raw relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2]];carrier Fin(6)

lexicographically sorted distinct ordered pairs=[[0, 0], [0, 2], [2, 2], [3, 3], [4, 4], [5, 5]]

carrier label=0;required diagonal=[0, 0]

present=Yes

carrier label=1;required diagonal=[1, 1]

present=No

carrier label=2;required diagonal=[2, 2]

present=Yes

carrier label=3;required diagonal=[3, 3]

present=Yes

carrier label=4;required diagonal=[4, 4]

present=Yes

carrier label=5;required diagonal=[5, 5]

present=Yes

Reverse checks

• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(6, [1], No)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.