OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 10276: Validity of a Finite Total-Function Graph

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 52

← Question derivation index · Original question batch

R0019-10276 · q-20261003-r06-function-graph-0076

Original question and givens

The declared domain is D=∅ and codomain is B=∅. Is the ordered-pair set ∅ exactly the graph of a total function D→B?

  • G001 — The declared domain is D=∅ and codomain is B=∅. Is the ordered-pair set ∅ 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.

Step-by-step derivation

S001 · implies

Before

raw domain=[]

After

sorted distinct declared domain=[]

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 dependencies

Rule: function_graph_declared_domain_scope · Uses: G001

S002 · implies

Before

codomain=[];raw pair rows=[]

After

ordered-pair set=[]

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 dependencies

Rule: graph_set_duplicate_pairs_count_once · Uses: G001, S001

S003 · implies

Before

all pair type flags=[];source fiber sizes=[]

After

total function graph=Yes

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 dependencies

Rule: total_graph_type_and_totality_joint_gate · 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

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 83a4b7dbadb4bc53f6fe3354ef83d2b9237c222d4a1311d022481110cd1ecde6
  • candidate record SHA-256: 16382fcf47d4ba6c9c744e2f713e43ea59aa92b731a317297a29b0a593c4971f
  • semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5
  • question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON