Question 2872: Complete Codomain Fiber Profile

Original question and givens

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (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={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (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=[-6, -5, -4, -3, -2, -1, 0]; B=[0, 1, 2, 3, 4, 5, 6]; table=[[-6, 1], [-5, 2], [-4, 3], [-3, 4], [-2, 5], [-1, 6], [0, 0]]

legal total function on the declared A→B

input -6

f(-6)=1∈B; exactly one listed value

input -5

f(-5)=2∈B; exactly one listed value

input -4

f(-4)=3∈B; exactly one listed value

input -3

f(-3)=4∈B; exactly one listed value

input -2

f(-2)=5∈B; exactly one listed value

input -1

f(-1)=6∈B; exactly one listed value

input 0

f(0)=0∈B; exactly one listed value

target 0; all source rows examined

fiber=[0]; size=1

target 1; all source rows examined

fiber=[-6]; size=1

target 2; all source rows examined

fiber=[-5]; size=1

target 3; all source rows examined

fiber=[-4]; size=1

target 4; all source rows examined

fiber=[-3]; size=1

target 5; all source rows examined

fiber=[-2]; size=1

target 6; all source rows examined

fiber=[-1]; size=1

[]

all fibers have size≤1: Yes

missing codomain targets=[]

all codomain fibers nonempty: Yes

injective=Yes;surjective=Yes

requested result={(0, 1), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1)}

Reverse checks

• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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, 1), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 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.