# Question 35013: Ordered-Pair Membership in a Binary Relation [Return to catalogue page 176](https://ocme.evemisslab.com/en/questions/page/176/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-advance-20261004-r03-relation-foundations/#q-20261003-advance-20261004-r03-relation-0013) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0069-35013 · q-20261003-advance-20261004-r03-relation-0013 ## Original question and givens The binary relation has source A={0, 1, 2}, target B={4, 5, 6}, and R=∅⊆A×B. Does the ordered pair (0, 4) belong to R? - G001 — The binary relation has source A={0, 1, 2}, target B={4, 5, 6}, and R=∅⊆A×B. Does the ordered pair (0, 4) belong to R? ## 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-relation](https://ocme.evemisslab.com/en/mko/mko-relation/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-relation-pair-membership, [A, B, R, operation, x, y]] After derived = [pair_membership, no-invented-domain] Conditions: none Keep EACH bank’s keys/operation/domain distinct:LDL/PULSE/function legacy table underscores,TREE full tree_* with its declared domain,and permissive BASE signed without invented op/domain.All arithmetic respects the binary-primitive contract. Step identifiers and dependenciesRule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001 S002 · impliesBefore source roles = [relation A ORIGINAL, 100, [0, 1, 2]] After derived = [[0, 1, 2]] Conditions: none Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits. Step identifiers and dependenciesRule: Finite_RAW_ALL_SAFE_distinct_labels_then_derived_sort · Uses: G001, S001 S003 · impliesBefore source roles = [relation B ORIGINAL, 100, [4, 5, 6]] After derived = [[4, 5, 6]] Conditions: none Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits. Step identifiers and dependenciesRule: Finite_RAW_ALL_SAFE_distinct_labels_then_derived_sort · Uses: G001, S002 S004 · impliesBefore source roles = [[0, 1, 2], [4, 5, 6], []] After derived = [Yes] Conditions: none FIRST qualify the WHOLE R as a subset of A×B.A relation may be one-to-many/many-to-one and need not be total or a function.Typed raw duplicate pairs may normalize. Step identifiers and dependenciesRule: Relation_WHOLE_R_subset_declared_AcrossB_NO_function_premise · Uses: G001, S003 S005 · impliesBefore source roles = [0, 4] After derived = [[], No] Conditions: none Compare the COMPLETE ordered pair,not separately occurring coordinates.A query outside carriers or in empty R is a LEGAL No.Do not force queried labels into A/B or confuse(y,x). Step identifiers and dependenciesRule: Relation_PAIR_query_keep_component_order_OUTSIDE_is_legal_No · Uses: G001, S004 ## Reverse checks - S005 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged. - S004 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged. - S003 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged. - S002 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged. - S001 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged. ## Result 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: b31e3651d8db340390a22d3b14660dfd2b78bb467078823e6fac7d6ab6eec72c - candidate record SHA-256: 7650b3a17e52fcdeeef28ec1543792041977788b1a0f072a14129e0b27e96b7e - semantic review receipt SHA-256: ccf93c72638d6abf6a412514da5838d55ecb09b71d0e4fdec2b559dc9fdee14b - question-first witness SHA-256: 11af9a338b5a610342b2cfb2ee295683131b3139ac402dd19b1ba1725976d518 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-advance-20261004-r03-relation-0013.json)