Question 2870: Complete Codomain Fiber Profile

Original question and givens

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. 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={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. 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=[-5, -4, -3, -2, -1, 0]; B=[0, 1, 2, 3, 4]; table=[[-5, 0], [-4, 1], [-3, 2], [-2, 3], [-1, 4], [0, 0]]

legal total function on the declared A→B

input -5

f(-5)=0∈B; exactly one listed value

input -4

f(-4)=1∈B; exactly one listed value

input -3

f(-3)=2∈B; exactly one listed value

input -2

f(-2)=3∈B; exactly one listed value

input -1

f(-1)=4∈B; exactly one listed value

input 0

f(0)=0∈B; exactly one listed value

target 0; all source rows examined

fiber=[-5, 0]; size=2

target 1; all source rows examined

fiber=[-4]; size=1

target 2; all source rows examined

fiber=[-3]; size=1

target 3; all source rows examined

fiber=[-2]; size=1

target 4; all source rows examined

fiber=[-1]; size=1

[[0, [-5, 0]]]

all fibers have size≤1: No

missing codomain targets=[]

all codomain fibers nonempty: Yes

injective=No;surjective=Yes

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

Reverse checks

• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

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

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.