Question 14841: An Injection Certifies One Finite Cardinality Inequality

Original question and givens

Use the complete finite source=[150, 151, 152, 153, 154] and target=[250, 251] as distinct-element lists in their original orders, with mapping=[250, 251, 250, 251, 250] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

• G001 — Use the complete finite source=[150, 151, 152, 153, 154] and target=[250, 251] as distinct-element lists in their original orders, with mapping=[250, 251, 250, 251, 250] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

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-cardinality-comparison

Step-by-step derivation

exact ordered source=[150, 151, 152, 153, 154];exact ordered target=[250, 251];aligned mapping=[250, 251, 250, 251, 250]

total typed map on these distinct source/target lists;orders retained

distinct source indices=[0, 1];source labels=[150, 151];images=[250, 251]

same image=No

distinct source indices=[0, 2];source labels=[150, 152];images=[250, 250]

same image=Yes

distinct source indices=[0, 3];source labels=[150, 153];images=[250, 251]

same image=No

distinct source indices=[0, 4];source labels=[150, 154];images=[250, 250]

same image=Yes

distinct source indices=[1, 2];source labels=[151, 152];images=[251, 250]

same image=No

distinct source indices=[1, 3];source labels=[151, 153];images=[251, 251]

same image=Yes

distinct source indices=[1, 4];source labels=[151, 154];images=[251, 250]

same image=No

distinct source indices=[2, 3];source labels=[152, 153];images=[250, 251]

same image=No

distinct source indices=[2, 4];source labels=[152, 154];images=[250, 250]

same image=Yes

distinct source indices=[3, 4];source labels=[153, 154];images=[251, 250]

same image=No

target label=250;exact original mapping=[250, 251, 250, 251, 250]

preimage source indices=[0, 2, 4];covered=Yes

target label=251;exact original mapping=[250, 251, 250, 251, 250]

preimage source indices=[1, 3];covered=Yes

all source collision witnesses=[[0, 2, 250], [0, 4, 250], [1, 3, 251], [2, 4, 250]];injective=No

this mapping comparison certificate=no_comparison_certificate

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

([150, 151, 152, 153, 154], [250, 251], [250, 251, 250, 251, 250], No, no_comparison_certificate)

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.