Question 2653: Repeated Inputs versus a Single-Valued Pair Table

Original question and givens

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -10], [1, -9], [1, -9], [2, -1]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

• G001 — Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -10], [1, -9], [1, -9], [2, -1]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

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-mathematical-representation

Step-by-step derivation

declared symbolic rule or complete finite table

keep the stated domain and representation distinct

input=0; distinct outputs=[-10]

one output=Yes

input=1; distinct outputs=[-9]

one output=Yes

input=2; distinct outputs=[-1]

one output=Yes

[]

function on listed domain=Yes

Reverse checks

• 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, -10], [1, -9], [2, -1]], [0, 1, 2], [], Yes)

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.