# Question 20883: A Directed Edge Is an Ordered Pair under the Declared Simple Convention [Return to catalogue page 105](https://ocme.evemisslab.com/en/questions/page/105/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261108-r07-finite-directed-graph-foundations/#q-20261014-advance-20261108-r07-finite-directed-graph-0008) 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. R0040-20883 · q-20261014-advance-20261108-r07-finite-directed-graph-0008 ## Original question and givens Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention). - G001 — Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention). ## 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-finite-directed-graph](https://ocme.evemisslab.com/en/mko/mko-finite-directed-graph/) ## Step-by-step derivation S001 · impliesBefore source operation=digraph-qualification;original snapshot and mathematics retained After explicit ordered source output fields=[vertex_count, edges, loop_free, simple_directed_set_qualified, convention] Conditions: none Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL complete directed carrier Fin(5);ORIGINAL lexicographically canonical ORDERED arcs=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]] After all endpoints in range and same-direction arcs unique;actual loops=[];main loop-free directed-set convention qualified Conditions: none Check outer from/to order without sorting endpoints inside each arc.(u,v) and(v,u)are distinct and may coexist.The main convention forbids loops;the boundary temporarily accepts them to audit that convention,not to deny other digraph models. Step identifiers and dependenciesRule: digraph_ordered_pairs_not_undirected_endpoint_sorting · Uses: G001, S001 S003 · impliesBefore complete ordered-arc data passed this main loop-free simple-directed convention After qualification does NOT assert acyclicity,undirectedness,connectivity or any degree convention beyond this signature Conditions: none Opposite arcs or a directed cycle may occur in a qualified loop-free simple digraph.This true flag does not promote additional graph properties. Step identifiers and dependenciesRule: digraph_signature_qualification_not_unrequested_graph_properties · Uses: G001, S002 ## Reverse checks - 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 (5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges) 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: 715817f97d3afb6f787f660c124b2d268ffe4dada01f5f4ccd4b897fb629e53d - candidate record SHA-256: 2d14ad2b4d77cd9ec928eed9a8e83a4a56a523a5787be84f5c550f726c2c2d7f - semantic review receipt SHA-256: c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d - question-first witness SHA-256: e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0008.json)