# Question 8343: Typed Converse Relation and Double Converse [Return to catalogue page 42](https://ocme.evemisslab.com/en/questions/page/42/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261017-r03-relation-foundations/#q-20261012-advance-20261017-r03-relation-0068) 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. R0015-08343 · q-20261012-advance-20261017-r03-relation-0068 ## Original question and givens The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?). - G001 — The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?). ## 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 types=(3,3);raw ordered pairs=[[0, 0], [0, 2], [1, 0]] After normalized relation=[[0, 0], [0, 2], [1, 0]] Conditions: none Validate both types of every ordered pair,then deduplicate identical pairs.Swapping coordinates forms the converse,not a sorting of one pair.A relation may be multivalued or partial. Step identifiers and dependenciesRule: typed_relation_set_normalization · Uses: G001 S002 · impliesBefore phase=converse;pair=[0, 0] After swapped=[0, 0] Conditions: none Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function. Step identifiers and dependenciesRule: relation_swap_ordered_coordinates · Uses: G001, S001 S003 · impliesBefore phase=converse;pair=[0, 2] After swapped=[2, 0] Conditions: none Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function. Step identifiers and dependenciesRule: relation_swap_ordered_coordinates · Uses: G001, S002 S004 · impliesBefore phase=converse;pair=[1, 0] After swapped=[0, 1] Conditions: none Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function. Step identifiers and dependenciesRule: relation_swap_ordered_coordinates · Uses: G001, S003 S005 · impliesBefore phase=double converse;pair=[0, 0] After swapped=[0, 0] Conditions: none Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function. Step identifiers and dependenciesRule: relation_swap_ordered_coordinates · Uses: G001, S004 S006 · impliesBefore phase=double converse;pair=[0, 1] After swapped=[1, 0] Conditions: none Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function. Step identifiers and dependenciesRule: relation_swap_ordered_coordinates · Uses: G001, S005 S007 · impliesBefore phase=double converse;pair=[2, 0] After swapped=[0, 2] Conditions: none Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function. Step identifiers and dependenciesRule: relation_swap_ordered_coordinates · Uses: G001, S006 S008 · impliesBefore old types=[3, 3];converse types=[3, 3];converse=[[0, 0], [0, 1], [2, 0]] After double converse=[[0, 0], [0, 2], [1, 0]];equal=Yes Conditions: none Swapping twice returns the original types and pair set even when the relation is not a function. Step identifiers and dependenciesRule: double_converse_typed_audit · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([3, 3], [3, 3], [[0, 0], [0, 2], [1, 0]], [[0, 0], [0, 1], [2, 0]], [[0, 0], [0, 2], [1, 0]], Yes) 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: 50a805c6b5d522071b248d5d6e8c465f2e98e457f2e521e5f31275e9e2a35fb9 - candidate record SHA-256: 76debd4636f380e6576687e2926a1f862bc76da74c65e4e44c0843d465adb361 - semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18 - question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r03-relation-0068.json)