Question 8326: Typed Converse Relation and Double Converse

Original question and givens

The explicitly typed relation R⊆Fin(0)×Fin(0) has ordered pairs=[]. 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(0)×Fin(0) has ordered pairs=[]. 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=(0,0);raw ordered pairs=[]

normalized relation=[]

old types=[0, 0];converse types=[0, 0];converse=[]

double converse=[];equal=Yes

Reverse checks

• 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

([0, 0], [0, 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.