Question 8345: Typed Converse Relation and Double Converse

Original question and givens

The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [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, 1], [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

Step-by-step derivation

types=(3,3);raw ordered pairs=[[0, 0], [0, 1], [0, 2], [1, 0]]

normalized relation=[[0, 0], [0, 1], [0, 2], [1, 0]]

phase=converse;pair=[0, 0]

swapped=[0, 0]

phase=converse;pair=[0, 1]

swapped=[1, 0]

phase=converse;pair=[0, 2]

swapped=[2, 0]

phase=converse;pair=[1, 0]

swapped=[0, 1]

phase=double converse;pair=[0, 0]

swapped=[0, 0]

phase=double converse;pair=[0, 1]

swapped=[1, 0]

phase=double converse;pair=[1, 0]

swapped=[0, 1]

phase=double converse;pair=[2, 0]

swapped=[0, 2]

old types=[3, 3];converse types=[3, 3];converse=[[0, 0], [0, 1], [1, 0], [2, 0]]

double converse=[[0, 0], [0, 1], [0, 2], [1, 0]];equal=Yes

Reverse checks

• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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, 1], [0, 2], [1, 0]], [[0, 0], [0, 1], [1, 0], [2, 0]], [[0, 0], [0, 1], [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.