Question 34954: Converse of a Binary Relation

On this page

Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

← Question derivation index · Original question batch

R0068-34954 · q-20261003-advance-20261004-r03-relation-0079

Original question and givens

The binary relation has source A={-9, -8, -7}, target B={-5, -4, -3}, and R={(-9, -4), (-8, -4)}⊆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={-9, -8, -7}, target B={-5, -4, -3}, and R={(-9, -4), (-8, -4)}⊆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.

mko-relation

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, [-9, -8, -7]]

After

derived = [[-9, -8, -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.

Step identifiers and dependencies

Rule: LDL_RAW_distinct_SAFE_set_empty_allowed · Uses: G001, S001

S003 · implies

Before

source roles = [relation ORIGINAL target B, [-5, -4, -3]]

After

derived = [[-5, -4, -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.

Step identifiers and dependencies

Rule: LDL_RAW_distinct_SAFE_set_empty_allowed · Uses: G001, S002

S004 · implies

Before

source roles = [0, [-9, -4]]

After

derived = [[-9, -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 dependencies

Rule: Relation_RAW_EVERY_pair_typed_BEFORE_dedup · Uses: G001, S003

S005 · implies

Before

source roles = [1, [-8, -4]]

After

derived = [[-8, -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 dependencies

Rule: Relation_RAW_EVERY_pair_typed_BEFORE_dedup · Uses: G001, S004

S006 · implies

Before

source roles = [[-9, -8, -7], [-5, -4, -3], [[-9, -4], [-8, -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 dependencies

Rule: Relation_COMPLETE_typed_subset_A_cross_B_NO_function_assumption · Uses: G001, S005

S007 · implies

Before

source roles = [-9, -4, A x B]

After

derived = [-4, -9, 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, S006

S008 · implies

Before

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

After

derived = [-4, -8, 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, -4, -3], [-9, -8, -7], [[-4, -9], [-4, -8]]]

After

derived = [{(-4, -9), (-4, -8)}]

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, S008

Reverse checks

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

{(-4, -9), (-4, -8)}

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: 7820f592214397b925248f579df1c220b58b08cb72d578fa5a241695b64af1ae
candidate record SHA-256: 07b78ddb5fb6c27b22e1f144cde7629c8472f51aec383d32c2475edfd6d674e8
semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed
question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON