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R0068-34964 · q-20261003-advance-20261004-r03-relation-0089
The binary relation has source A={1, 2, 3}, target B={5, 6, 7}, and R={(1, 5)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.
G001 — The binary relation has source A={1, 2, 3}, target B={5, 6, 7}, and R={(1, 5)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.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.
Before
source roles = [qf-relation-converse-set, [A, B, R, operation]]
After
derived = [converse, 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.
Rule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001
Before
source roles = [relation ORIGINAL source A, [1, 2, 3]]
After
derived = [[1, 2, 3]]
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.
Rule: LDL_RAW_distinct_SAFE_set_empty_allowed · Uses: G001, S001
Before
source roles = [relation ORIGINAL target B, [5, 6, 7]]
After
derived = [[5, 6, 7]]
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.
Rule: LDL_RAW_distinct_SAFE_set_empty_allowed · Uses: G001, S002
Before
source roles = [0, [1, 5]]
After
derived = [[1, 5]]
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.
Rule: Relation_RAW_EVERY_pair_typed_BEFORE_dedup · Uses: G001, S003
Before
source roles = [[1, 2, 3], [5, 6, 7], [[1, 5]]]
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.
Rule: Relation_COMPLETE_typed_subset_A_cross_B_NO_function_assumption · Uses: G001, S004
Before
source roles = [1, 5, A x B]
After
derived = [5, 1, B x A]
Conditions: none
Swap EACH ORIGINAL(x,y)to(y,x),exchanging source/target roles too.This is not merely reversing the list order.
Rule: Relation_CONVERSE_EACH_pair_swaps_roles_and_carriers · Uses: G001, S005
Before
source roles = [[5, 6, 7], [1, 2, 3], [[5, 1]]]
After
derived = [{(5, 1)}]Conditions: none
Display in NUMERIC lexicographic order by first,then second coordinate,without duplicates.R⁻¹ is a converse RELATION,not automatically an inverse FUNCTION.Empty R stays empty;do not invent a bijection premise.
Rule: Relation_CONVERSE_numeric_lex_order_not_inverse_FUNCTION · Uses: G001, S006
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.{(5, 1)}The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: b40c17d8784e4c8552828c2e84a3097c7f6d63a0d3aef2c548f8616c9f7f66cbcandidate record SHA-256: d9d6253404b0e7ec455ec15624fba5df3665cfe06a7ffcdf938df981d8b591a5semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066edquestion-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229Lean: NOT_RUN · human review: NOT_PERFORMED