Question 2863: Complete Codomain Fiber Profile

Original question and givens

The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

• G001 — The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

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-bijective

Step-by-step derivation

A=[-3, -2, -1, 0]; B=[-5, -4, -3, -2, -1]; table=[[-3, -1], [-2, -1], [-1, -2], [0, -2]]

legal total function on the declared A→B

input -3

f(-3)=-1∈B; exactly one listed value

input -2

f(-2)=-1∈B; exactly one listed value

input -1

f(-1)=-2∈B; exactly one listed value

input 0

f(0)=-2∈B; exactly one listed value

target -5; all source rows examined

fiber=[]; size=0

target -4; all source rows examined

fiber=[]; size=0

target -3; all source rows examined

fiber=[]; size=0

target -2; all source rows examined

fiber=[-1, 0]; size=2

target -1; all source rows examined

fiber=[-3, -2]; size=2

[[-2, [-1, 0]], [-1, [-3, -2]]]

all fibers have size≤1: No

missing codomain targets=[-5, -4, -3]

all codomain fibers nonempty: No

injective=No;surjective=No

requested result={(-5, 0), (-4, 0), (-3, 0), (-2, 2), (-1, 2)}

Reverse checks

• 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

{(-5, 0), (-4, 0), (-3, 0), (-2, 2), (-1, 2)}

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.