Question 5596: Actual Range and Unused Codomain

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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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R0010-05596 · q-20261008-advance-20261012-r07-function-mapping-0046

Original question and givens

The complete total map f:Fin(2)→Fin(8) has images=[1, 4] indexed from 0; Fin(2)={0,…,1}. Every table entry must lie in the target type. Give (increasing actual range, increasing unused codomain elements, does the range equal the entire codomain?).

G001 — The complete total map f:Fin(2)→Fin(8) has images=[1, 4] indexed from 0; Fin(2)={0,…,1}. Every table entry must lie in the target type. Give (increasing actual range, increasing unused codomain elements, does the range equal the entire codomain?).

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-mapping

Step-by-step derivation

S001 · implies

Before

f:Fin(2)→Fin(8);source indices start at0

After

complete total map,not a partial program function

Conditions: none

Every legal source element has one codomain value.Fin(0) is empty;an outside query is ill-typed,not a hole of a partial function at a legal input.

Step identifiers and dependencies

Rule: total_finite_mapping_scope · Uses: G001

S002 · implies

Before

f input=0∈Fin(2)

After

f(0)=1∈Fin(8)

Conditions: none

Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal.

Step identifiers and dependencies

Rule: complete_mapping_entry · Uses: G001, S001

S003 · implies

Before

f input=1∈Fin(2)

After

f(1)=4∈Fin(8)

Conditions: none

Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal.

Step identifiers and dependencies

Rule: complete_mapping_entry · Uses: G001, S002

S004 · implies

Before

codomain element=0

After

preimages=[];used=No

Conditions: none

Audit every codomain target;an unhit target remains in the codomain but not in the actual range.

Step identifiers and dependencies

Rule: codomain_fiber_inventory · Uses: G001, S003

S005 · implies

Before

codomain element=1

After

preimages=[0];used=Yes

Conditions: none

Audit every codomain target;an unhit target remains in the codomain but not in the actual range.

Step identifiers and dependencies

Rule: codomain_fiber_inventory · Uses: G001, S004

S006 · implies

Before

codomain element=2

After

preimages=[];used=No

Conditions: none

Audit every codomain target;an unhit target remains in the codomain but not in the actual range.

Step identifiers and dependencies

Rule: codomain_fiber_inventory · Uses: G001, S005

S007 · implies

Before

codomain element=3

After

preimages=[];used=No

Conditions: none

Audit every codomain target;an unhit target remains in the codomain but not in the actual range.

Step identifiers and dependencies

Rule: codomain_fiber_inventory · Uses: G001, S006

S008 · implies

Before

codomain element=4

After

preimages=[1];used=Yes

Conditions: none

Audit every codomain target;an unhit target remains in the codomain but not in the actual range.

Step identifiers and dependencies

Rule: codomain_fiber_inventory · Uses: G001, S007

S009 · implies

Before

codomain element=5

After

preimages=[];used=No

Conditions: none

Audit every codomain target;an unhit target remains in the codomain but not in the actual range.

Step identifiers and dependencies

Rule: codomain_fiber_inventory · Uses: G001, S008

S010 · implies

Before

codomain element=6

After

preimages=[];used=No

Conditions: none

Audit every codomain target;an unhit target remains in the codomain but not in the actual range.

Step identifiers and dependencies

Rule: codomain_fiber_inventory · Uses: G001, S009

S011 · implies

Before

codomain element=7

After

preimages=[];used=No

Conditions: none

Audit every codomain target;an unhit target remains in the codomain but not in the actual range.

Step identifiers and dependencies

Rule: codomain_fiber_inventory · Uses: G001, S010

S012 · implies

Before

actual range=[1, 4];unused codomain=[0, 2, 3, 5, 6, 7]

After

range equals codomain=No

Conditions: none

Totality does not imply surjectivity;the range equals the codomain only if every target has a preimage.The empty map Fin(0)→Fin(0) satisfies this criterion.

Step identifiers and dependencies

Rule: range_vs_declared_codomain · Uses: G001, S011

Reverse checks

S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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

([1, 4], [0, 2, 3, 5, 6, 7], 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: 2d77e1db386c980d7bc164c47e3769896d10e3b359d3ba0a0d9472bab83e13af
candidate record SHA-256: 32474edaae626768c21b0179dbf6cf7b3ed27cce3e34554739f598ad673199f4
semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f
question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc
Lean: NOT_RUN · human review: NOT_PERFORMED

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