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Question 10282: 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

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R0019-10282 · q-20261003-r06-function-graph-0082

Original question and givens

The declared domain is D={-12, -11} and codomain is B={-4, -3, -2}. Is the ordered-pair set {(-12, -4)} exactly the graph of a total function D→B?

  • G001 — The declared domain is D={-12, -11} and codomain is B={-4, -3, -2}. Is the ordered-pair set {(-12, -4)} 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=[-12, -11]

After

sorted distinct declared domain=[-12, -11]

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=[-4, -3, -2];raw pair rows=[[-12, -4]]

After

ordered-pair set=[[-12, -4]]

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

pair=[-12, -4];domain=[-12, -11];codomain=[-4, -3, -2]

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 dependencies

Rule: graph_every_pair_type_guard · Uses: G001, S002

S004 · implies

Before

declared input=-12;normalized pair set=[[-12, -4]]

After

distinct output fiber=[-4];count=1

Conditions: none

Each declared input needs exactly one distinct output.At most one point gives single-valuedness,not totality.

Step identifiers and dependencies

Rule: graph_each_declared_input_exactly_one · Uses: G001, S003

S005 · implies

Before

declared input=-11;normalized pair set=[[-12, -4]]

After

distinct output fiber=[];count=0

Conditions: none

Each declared input needs exactly one distinct output.At most one point gives single-valuedness,not totality.

Step identifiers and dependencies

Rule: graph_each_declared_input_exactly_one · Uses: G001, S004

S006 · implies

Before

all pair type flags=[Yes];source fiber sizes=[1, 0]

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 dependencies

Rule: total_graph_type_and_totality_joint_gate · Uses: G001, S005

Reverse checks

  • 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: ecf044b20e15e6f99486f7de3a8b215d9cbda71dc92a7293f6637aad8c6a7d98
  • candidate record SHA-256: a7bebe76ea12d37a5ab0a1f37fe0239efd8c3f5159aecebe5db126c89d0daffe
  • semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5
  • question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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