# Question 35232: Edge Membership in a Simple Undirected Graph [Return to catalogue page 177](https://ocme.evemisslab.com/en/questions/page/177/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-advance-20261004-r04-graph-foundations/#q-20261003-advance-20261004-r04-graph-0007) 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. R0069-35232 · q-20261003-advance-20261004-r04-graph-0007 ## Original question and givens The finite simple undirected graph has V={-6, -5, -4, -3, -2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-2, -6} an edge in E? - G001 — The finite simple undirected graph has V={-6, -5, -4, -3, -2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-2, -6} an edge in E? ## 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-graph](https://ocme.evemisslab.com/en/mko/mko-graph/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-simple-graph-edge-membership, [E, V, operation, u, v]] After derived = [edge_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 = [LDL graph vertices ORIGINAL, 100, [-6, -5, -4, -3, -2]] After derived = [[-6, -5, -4, -3, -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 = [[-6, -5, -4, -3, -2], [], No] After derived = [[], [[-6, []], [-5, []], [-4, []], [-3, []], [-2, []]]] Conditions: none Type the WHOLE raw edge list,then normalize undirected endpoints/deduplicate;reversed duplicates are allowed.INPUT edges have no loops/outside endpoints,while isolated vertices stay in V.TREE raw E does NOT acquire LDL’s200-entry limit. Step identifiers and dependenciesRule: Graph_WHOLE_simple_undirected_edges_normalized_NO_loops_or_outside · Uses: G001, S002 S004 · impliesBefore source roles = [-2, -6] After derived = [[-6, -2], No] Conditions: none An undirected QUERY ignores endpoint order.A same-endpoint or outside-vertex query is a LEGAL No;do not confuse it with INPUT edge qualification. Step identifiers and dependenciesRule: Graph_EDGE_query_outside_or_same_endpoint_LEGAL_No · Uses: G001, S003 ## Reverse checks - 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: 765ce566d091cd3b9f02a4d83a8f8df4f7eb97e88c8aeca167e7034a668cc996 - candidate record SHA-256: e78a9597123c073514dcdc9782c1f780a639723b68d2d16387231c3085b858ff - 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-r04-graph-0007.json)