# Question 10294: Validity of a Finite Total-Function Graph [Return to catalogue page 52](https://ocme.evemisslab.com/en/questions/page/52/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-r06-function-graph-foundations/#q-20261003-r06-function-graph-0094) 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. R0019-10294 · q-20261003-r06-function-graph-0094 ## Original question and givens The declared domain is D={24, 25} and codomain is B={8, 9, 10}. Is the ordered-pair set {(24, 8), (25, 48)} exactly the graph of a total function D→B? - G001 — The declared domain is D={24, 25} and codomain is B={8, 9, 10}. Is the ordered-pair set {(24, 8), (25, 48)} exactly the graph of a total function D→B? ## 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-function-graph](https://ocme.evemisslab.com/en/mko/mko-function-graph/) ## Step-by-step derivation S001 · impliesBefore raw domain=[24, 25] After sorted distinct declared domain=[24, 25] Conditions: none The domain is the declared finite integer subset,not automatically all R.The input list must already be distinct.The first coordinate is input,the second output. Step identifiers and dependenciesRule: function_graph_declared_domain_scope · Uses: G001 S002 · impliesBefore codomain=[8, 9, 10];raw pair rows=[[24, 8], [25, 48]] After ordered-pair set=[[24, 8], [25, 48]] Conditions: none This protocol treats the graph as a set:deduplicate identical pairs,but retain different y values at one x.This differs from a raw-row protocol requiring one original row per input. Step identifiers and dependenciesRule: graph_set_duplicate_pairs_count_once · Uses: G001, S001 S003 · impliesBefore pair=[24, 8];domain=[24, 25];codomain=[8, 9, 10] After both coordinates typed=Yes Conditions: none Every pair must lie in D×B.Do not ignore outside-domain or outside-codomain pairs after finding legal points for some inputs. Step identifiers and dependenciesRule: graph_every_pair_type_guard · Uses: G001, S002 S004 · impliesBefore pair=[25, 48];domain=[24, 25];codomain=[8, 9, 10] After both coordinates typed=No Conditions: none Every pair must lie in D×B.Do not ignore outside-domain or outside-codomain pairs after finding legal points for some inputs. Step identifiers and dependenciesRule: graph_every_pair_type_guard · Uses: G001, S003 S005 · impliesBefore declared input=24;normalized pair set=[[24, 8], [25, 48]] After distinct output fiber=[8];count=1 Conditions: none Each declared input needs exactly one distinct output.At most one point gives single-valuedness,not totality. Step identifiers and dependenciesRule: graph_each_declared_input_exactly_one · Uses: G001, S004 S006 · impliesBefore declared input=25;normalized pair set=[[24, 8], [25, 48]] After distinct output fiber=[48];count=1 Conditions: none Each declared input needs exactly one distinct output.At most one point gives single-valuedness,not totality. Step identifiers and dependenciesRule: graph_each_declared_input_exactly_one · Uses: G001, S005 S007 · impliesBefore all pair type flags=[Yes, No];source fiber sizes=[1, 1] After total function graph=No Conditions: none Typing and totality are joint conditions.An empty domain with an empty graph is valid,but extra outside pairs are not.The valid empty function does not require a nonempty codomain. Step identifiers and dependenciesRule: total_graph_type_and_totality_joint_gate · Uses: G001, S006 ## Reverse checks - 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 No 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: 3d4fc0cad72f34e91594819cba96cdfc7cc7d772ca9f5fd0175cd29851454cdd - candidate record SHA-256: b89b958d6af5511e7174adea8853c43730f4cca1d1b3525a9498e8a5c6468c11 - semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5 - question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-r06-function-graph-0094.json)