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Question 34967: Converse of a Binary Relation

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0068-34967 · q-20261003-advance-20261004-r03-relation-0092

Original question and givens

The binary relation has source A={4, 5, 6}, target B={8, 9, 10}, and R={(4, 8), (5, 9), (6, 10)}⊆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={4, 5, 6}, target B={8, 9, 10}, and R={(4, 8), (5, 9), (6, 10)}⊆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.

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.

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001

S002 · implies

Before

source roles = [relation ORIGINAL source A, [4, 5, 6]]

After

derived = [[4, 5, 6]]

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 dependencies

Rule: LDL_RAW_distinct_SAFE_set_empty_allowed · Uses: G001, S001

S003 · implies

Before

source roles = [relation ORIGINAL target B, [8, 9, 10]]

After

derived = [[8, 9, 10]]

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 dependencies

Rule: LDL_RAW_distinct_SAFE_set_empty_allowed · Uses: G001, S002

S004 · implies

Before

source roles = [0, [4, 8]]

After

derived = [[4, 8]]

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 dependencies

Rule: Relation_RAW_EVERY_pair_typed_BEFORE_dedup · Uses: G001, S003

S005 · implies

Before

source roles = [1, [5, 9]]

After

derived = [[5, 9]]

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 dependencies

Rule: Relation_RAW_EVERY_pair_typed_BEFORE_dedup · Uses: G001, S004

S006 · implies

Before

source roles = [2, [6, 10]]

After

derived = [[6, 10]]

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 dependencies

Rule: Relation_RAW_EVERY_pair_typed_BEFORE_dedup · Uses: G001, S005

S007 · implies

Before

source roles = [[4, 5, 6], [8, 9, 10], [[4, 8], [5, 9], [6, 10]]]

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 dependencies

Rule: Relation_COMPLETE_typed_subset_A_cross_B_NO_function_assumption · Uses: G001, S006

S008 · implies

Before

source roles = [4, 8, A x B]

After

derived = [8, 4, 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.

Step identifiers and dependencies

Rule: Relation_CONVERSE_EACH_pair_swaps_roles_and_carriers · Uses: G001, S007

S009 · implies

Before

source roles = [5, 9, A x B]

After

derived = [9, 5, 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.

Step identifiers and dependencies

Rule: Relation_CONVERSE_EACH_pair_swaps_roles_and_carriers · Uses: G001, S008

S010 · implies

Before

source roles = [6, 10, A x B]

After

derived = [10, 6, 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.

Step identifiers and dependencies

Rule: Relation_CONVERSE_EACH_pair_swaps_roles_and_carriers · Uses: G001, S009

S011 · implies

Before

source roles = [[8, 9, 10], [4, 5, 6], [[8, 4], [9, 5], [10, 6]]]

After

derived = [{(8, 4), (9, 5), (10, 6)}]

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.

Step identifiers and dependencies

Rule: Relation_CONVERSE_numeric_lex_order_not_inverse_FUNCTION · Uses: G001, S010

Reverse checks

  • S011 · 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.
  • S010 · 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.
  • S009 · 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.
  • S008 · 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.
  • 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

{(8, 4), (9, 5), (10, 6)}

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: 6eb31fca665c48aead60fff0310f680f35929ef9193d69782bb6433668083649
  • candidate record SHA-256: d8f51f743e1fcb1c9fee88d692ff4344e5afe556690855fcc4279c911ba735e6
  • semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed
  • question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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