Question 10278: Validity of a Finite Total-Function Graph

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0019-10278 · q-20261003-r06-function-graph-0078

Original question and givens

The declared domain is D={-24, -23} and codomain is B={-8, -7, -6}. Is the ordered-pair set {(-24, -8), (-23, -7), (-24, -6)} exactly the graph of a total function D→B?

G001 — The declared domain is D={-24, -23} and codomain is B={-8, -7, -6}. Is the ordered-pair set {(-24, -8), (-23, -7), (-24, -6)} 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

Step-by-step derivation

S001 · implies

Before

raw domain=[-24, -23]

After

sorted distinct declared domain=[-24, -23]

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=[-8, -7, -6];raw pair rows=[[-24, -8], [-23, -7], [-24, -6]]

After

ordered-pair set=[[-24, -8], [-24, -6], [-23, -7]]

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=[-24, -8];domain=[-24, -23];codomain=[-8, -7, -6]

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

pair=[-24, -6];domain=[-24, -23];codomain=[-8, -7, -6]

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, S003

S005 · implies

Before

pair=[-23, -7];domain=[-24, -23];codomain=[-8, -7, -6]

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, S004

S006 · implies

Before

declared input=-24;normalized pair set=[[-24, -8], [-24, -6], [-23, -7]]

After

distinct output fiber=[-8, -6];count=2

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, S005

S007 · implies

Before

declared input=-23;normalized pair set=[[-24, -8], [-24, -6], [-23, -7]]

After

distinct output fiber=[-7];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, S006

S008 · implies

Before

all pair type flags=[Yes, Yes, Yes];source fiber sizes=[2, 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 dependencies

Rule: total_graph_type_and_totality_joint_gate · Uses: G001, S007

Reverse checks

S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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: bbcc6e359ad6ef2a12f56cca07335f2fb45beba1fb0d969b8069c4938c015ae5
candidate record SHA-256: 4e828e6db48d8118e0c254a5aac8178b0d831665334f7dd6ccfb48862feae4cd
semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5
question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419
Lean: NOT_RUN · human review: NOT_PERFORMED

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