# Question 34987: First Projection of a Binary Relation [Return to catalogue page 175](https://ocme.evemisslab.com/en/questions/page/175/) [← 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-0037) 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. R0068-34987 · q-20261003-advance-20261004-r03-relation-0037 ## Original question and givens The binary relation has source A={-1, 0, 1}, target B={3, 4, 5}, and R={(0, 3), (1, 4)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once. - G001 — The binary relation has source A={-1, 0, 1}, target B={3, 4, 5}, and R={(0, 3), (1, 4)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once. ## 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-domain-projection, [A, B, R, operation]] After derived = [domain_projection, no-invented-domain] Conditions: none Separate source banks:legacy RAW SAFE with unbounded derived BigInts versus BASE Number.isInteger without invented SAFE/million caps.Operations,domains and permissive/exact keys follow THIS bank.Keep ORIGINAL JSON intact. Step identifiers and dependenciesRule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001 S002 · impliesBefore source roles = [relation ORIGINAL source A, [-1, 0, 1]] After derived = [[-1, 0, 1]] Conditions: none THIS LDL set has0..100 DISTINCT RAW SAFE labels,with the empty set allowed.Sort only the derived representation;preserve original JSON order.Do not import another BASE set protocol. Step identifiers and dependenciesRule: LDL_RAW_distinct_SAFE_set_empty_allowed · Uses: G001, S001 S003 · impliesBefore source roles = [relation ORIGINAL target B, [3, 4, 5]] After derived = [[3, 4, 5]] Conditions: none THIS LDL set has0..100 DISTINCT RAW SAFE labels,with the empty set allowed.Sort only the derived representation;preserve original JSON order.Do not import another BASE set protocol. Step identifiers and dependenciesRule: LDL_RAW_distinct_SAFE_set_empty_allowed · Uses: G001, S002 S004 · impliesBefore source roles = [0, [0, 3]] After derived = [[0, 3]] Conditions: none Type EVERY raw R pair as EXACTLY TWO SAFE coordinates BEFORE deduplication.Identical repeated pairs are legal,but deduplication must not hide an unvalidated cell. Step identifiers and dependenciesRule: Relation_RAW_EVERY_pair_typed_BEFORE_dedup · Uses: G001, S003 S005 · impliesBefore source roles = [1, [1, 4]] After derived = [[1, 4]] Conditions: none Type EVERY raw R pair as EXACTLY TWO SAFE coordinates BEFORE deduplication.Identical repeated pairs are legal,but deduplication must not hide an unvalidated cell. Step identifiers and dependenciesRule: Relation_RAW_EVERY_pair_typed_BEFORE_dedup · Uses: G001, S004 S006 · impliesBefore source roles = [[-1, 0, 1], [3, 4, 5], [[0, 3], [1, 4]]] After derived = [Yes] Conditions: none FIRST qualify the COMPLETE R⊆A×B.A relation need not be total,single-valued,injective or onto.Empty R/carriers are allowed when typed.Keep ORIGINAL raw duplicates/order in the source copy. Step identifiers and dependenciesRule: Relation_COMPLETE_typed_subset_A_cross_B_NO_function_assumption · Uses: G001, S005 S007 · impliesBefore source roles = [[-1, 0, 1], [3, 4, 5], [[0, 3], [1, 4]]] After derived = [[0, 1], {0, 1}] Conditions: none Dom(R)collects ONLY first coordinates ACTUALLY appearing in pairs,once each.A may contain unused elements,so do not simply return A.Empty R has empty domain. Step identifiers and dependenciesRule: Relation_DOMAIN_actual_FIRST_coordinates_not_declared_A · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S006 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S005 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S004 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S003 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S002 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S001 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. ## Result {0, 1} 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: ec3ebb1bb9d9da954ed9b98c19855f8ff522b7186bddebb2b3812e54bccd0dec - candidate record SHA-256: 0a11e7c18e2c126586207dd5298346ab096c5f8260149a358d3ee2ad31a09fc5 - semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed - question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229 - 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-0037.json)