Question 32248: Reading the Domain of a Function

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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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R0063-32248 · q-20260930-r03-function-domain-set-0023

Original question and givens

Given the explicit declaration A={10, 11, 12}, B={-44, 0, 45}, ordered pairs=[(10, 0), (11, 45), (12, -44)] and a total function f:A→B, list the domain in increasing order.

G001 — Given the explicit declaration A={10, 11, 12}, B={-44, 0, 45}, ordered pairs=[(10, 0), (11, 45), (12, -44)] and a total function f:A→B, list the domain in increasing order.

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-domain-codomain-range

Step-by-step derivation

S001 · implies

Before

source roles = [qf-function-domain-set, domain, no-required-domain, []]

After

derived = [[declared_domain_set_STRING]]

Conditions: none

BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds.

Step identifiers and dependencies

Rule: R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle · Uses: G001

S002 · implies

Before

source roles = [A declared domain, pulse-function, [10, 11, 12]]

After

derived = [[10, 11, 12]]

Conditions: none

RAW A/B/C are distinct SAFE integers.Legacy injective limits100,but PULSE has NO such cap.Sort ONLY a derived work table,retaining original lists.

Step identifiers and dependencies

Rule: Map_RAW_distinct_SAFE_set_BANK_specific_cardinality_bound · Uses: G001, S001

S003 · implies

Before

source roles = [B declared codomain, pulse-function, [-44, 0, 45]]

After

derived = [[-44, 0, 45]]

Conditions: none

RAW A/B/C are distinct SAFE integers.Legacy injective limits100,but PULSE has NO such cap.Sort ONLY a derived work table,retaining original lists.

Step identifiers and dependencies

Rule: Map_RAW_distinct_SAFE_set_BANK_specific_cardinality_bound · Uses: G001, S002

S004 · implies

Before

source roles = [pulse-function, [[10, 0], [11, 45], [12, -44]]]

After

derived = [[[10, 0], [11, 45], [12, -44]]]

Conditions: none

PULSE rejects EXACT duplicate RAW rows BEFORE totality,not as a legal No.Legacy injective FIRST checks raw length≤200,then normalizes identical rows as one pair.Do not exchange these policies;type-check EVERY raw cell first.

Step identifiers and dependencies

Rule: Map_RAW_pairs_arity_SAFE_DUPLICATE_policy_BEFORE_totality · Uses: G001, S003

S005 · implies

Before

source roles = [pulse-function, 0, 10, 0, [10, 11, 12], [-44, 0, 45]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S004

S006 · implies

Before

source roles = [pulse-function, 1, 11, 45, [10, 11, 12], [-44, 0, 45]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S005

S007 · implies

Before

source roles = [pulse-function, 2, 12, -44, [10, 11, 12], [-44, 0, 45]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S006

S008 · implies

Before

source roles = [pulse-function, 10, [[10, 0], [11, 45], [12, -44]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S007

S009 · implies

Before

source roles = [pulse-function, 11, [[10, 0], [11, 45], [12, -44]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S008

S010 · implies

Before

source roles = [pulse-function, 12, [[10, 0], [11, 45], [12, -44]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S009

S011 · implies

Before

source roles = [pulse-function, [10, 11, 12], [-44, 0, 45], [[10, 0], [11, 45], [12, -44]], Yes]

After

derived = [[Yes, Yes, Yes], [1, 1, 1], Yes]

Conditions: none

A total function needs BOTH every declared input AND every row qualified.PULSE validity may legally return No;other queries and legacy injective REQUIRE actual totality BEFORE lookup.Empty A/table can be valid;unattained B labels remain.

Step identifiers and dependencies

Rule: Map_TOTAL_all_declared_inputs_AND_ALL_rows_before_any_lookup · Uses: G001, S010

S012 · implies

Before

source roles = [domain, [10, 11, 12], [-44, 0, 45], [[10, 0], [11, 45], [12, -44]]]

After

derived = [[10, 11, 12], {10, 11, 12}, []]

Conditions: none

Read DECLARED A for domain and DECLARED B for codomain;ONLY the range collects attained outputs.Unattained B remains in the codomain.An empty-domain function has empty range but may have nonempty codomain.

Step identifiers and dependencies

Rule: Function_DECLARED_domain_codomain_VERSUS_ACTUAL_range · Uses: G001, S011

Reverse checks

S012 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S011 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S010 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S009 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S008 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S007 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S006 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S005 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S004 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S003 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S002 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S001 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

{10, 11, 12}

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: 6e5708ff583ccc9b492f6dc5c63da052aa3b10f8716c9337612d4f0508e2d2bd
candidate record SHA-256: 69196f6e3a234518fb3d720b2f341bb550c462164963cd56afa488500bee4594
semantic review receipt SHA-256: acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0
question-first witness SHA-256: eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88
Lean: NOT_RUN · human review: NOT_PERFORMED

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