# Question 8360: Relation Single-Valuedness versus Total-Function Qualification [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-0085) 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. R0015-08360 · q-20261012-advance-20261017-r03-relation-0085 ## Original question and givens The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2]]. 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], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function). - G001 — The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2]]. 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], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function). ## 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]] After normalized relation=[[0, 0], [0, 2]] 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 source=0;pairs=[[0, 0], [0, 2]] After outgoing values=[0, 2];count=2 Conditions: none Collect all distinct outgoing targets of every declared source;an absent source has an empty fiber. Step identifiers and dependenciesRule: relation_each_source_value_fiber · Uses: G001, S001 S003 · impliesBefore source=1;pairs=[[0, 0], [0, 2]] After outgoing values=[];count=0 Conditions: none Collect all distinct outgoing targets of every declared source;an absent source has an empty fiber. Step identifiers and dependenciesRule: relation_each_source_value_fiber · Uses: G001, S002 S004 · impliesBefore source=2;pairs=[[0, 0], [0, 2]] After outgoing values=[];count=0 Conditions: none Collect all distinct outgoing targets of every declared source;an absent source has an empty fiber. Step identifiers and dependenciesRule: relation_each_source_value_fiber · Uses: G001, S003 S005 · impliesBefore source fiber sizes=[2, 0, 0] After single-valued=No;exactly-one total=No;table/status=not a total function Conditions: none At most one value permits missing sources;exactly one makes a total function.On an empty source both universal conditions are true and the table is empty.Repeating a pair does not increase a fiber size. Step identifiers and dependenciesRule: partial_single_valued_not_total · Uses: G001, S004 ## Reverse checks - 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], [[0, 0], [0, 2]], [2, 0, 0], No, No, not-a-total-function) 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: f0ca72d4f8ad4bf1c9c70d593df9c82ca88965251ca623f81fa01b5d3b17cfb2 - candidate record SHA-256: c5ef1ca7d3049e4f1232445aa2c1d1d6c293e08077ef093494fff73a1808bf86 - 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-0085.json)