# Question 30967: Swap Two Output Labels and Recheck Bijectivity [Return to catalogue page 155](https://ocme.evemisslab.com/en/questions/page/155/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261017-r01-bijective-foundations/#q-20261012-advance-20261017-r01-bijective-0067) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0060-30967 · q-20261012-advance-20261017-r01-bijective-0067 ## Original question and givens Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?). - G001 — Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?). ## 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](https://ocme.evemisslab.com/en/mko/mko-bijective/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density17-bij-output-transposition, bij-output-transposition, Fin_total_table] After derived = [[original_images, next_images, ALL_next_fibers, original_bij, next_bij, flags_agree]] Conditions: none density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status. Step identifiers and dependenciesRule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001 S002 · impliesBefore source roles = [Bijection ORIGINAL complete table, 5, 3, [1, 1, 1, 0, 0]] After derived = [[1, 1, 1, 0, 0]] Conditions: none Require the WHOLE declared table length to equal its source size and EVERY cell to lie in Fin(m),including unused cells.Even an empty outer source cannot waive this complete table typing obligation. Step identifiers and dependenciesRule: Composition_WHOLE_typed_table_ALL_cells_including_unused · Uses: G001, S001 S003 · calculationBefore Bijection ORIGINAL reverse ALL fiber sizes through target0 := sum(0, 2) After Bijection ORIGINAL reverse ALL fiber sizes through target0 = 2 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [Bijection ORIGINAL, [1, 1, 1, 0, 0], 3, 0] After derived = [[3, 4], 2] Conditions: none For EVERY declared target list ALL ORIGINAL source preimages,including empty fibers,not merely attained values.Retain original labels/index order. Step identifiers and dependenciesRule: Bijection_EVERY_declared_target_COMPLETE_fiber_including_EMPTY · Uses: G001, S003 S005 · calculationBefore Bijection ORIGINAL reverse ALL fiber sizes through target1 := sum(2, 3) After Bijection ORIGINAL reverse ALL fiber sizes through target1 = 5 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [Bijection ORIGINAL, [1, 1, 1, 0, 0], 3, 1] After derived = [[0, 1, 2], 3] Conditions: none For EVERY declared target list ALL ORIGINAL source preimages,including empty fibers,not merely attained values.Retain original labels/index order. Step identifiers and dependenciesRule: Bijection_EVERY_declared_target_COMPLETE_fiber_including_EMPTY · Uses: G001, S005 S007 · calculationBefore Bijection ORIGINAL reverse ALL fiber sizes through target2 := sum(5, 0) After Bijection ORIGINAL reverse ALL fiber sizes through target2 = 5 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [Bijection ORIGINAL, [1, 1, 1, 0, 0], 3, 2] After derived = [[], 0] Conditions: none For EVERY declared target list ALL ORIGINAL source preimages,including empty fibers,not merely attained values.Retain original labels/index order. Step identifiers and dependenciesRule: Bijection_EVERY_declared_target_COMPLETE_fiber_including_EMPTY · Uses: G001, S007 S009 · impliesBefore source roles = [Bijection ORIGINAL, [1, 1, 1, 0, 0], 3] After derived = [[[3, 4], [0, 1, 2], []], [[0, 1], [0, 2], [1, 2], [3, 4]], [2], No, No, No, 5] Conditions: none A collision has xempty has all three;empty->nonempty only injectivity. Step identifiers and dependenciesRule: Bijection_ALL_collisions_GLOBAL_LEX_missing_targets_THREE_distinct_properties · Uses: G001, S008 S010 · impliesBefore source roles = [5, 3, 0, 2, 0, 1] After derived = [1] Conditions: none Apply the SAME codomain-label transposition to EVERY original source output,without reordering sources or changing only the first occurrence.Equal a,b is a legal no-op. Step identifiers and dependenciesRule: Bijection_OUTPUT_label_transposition_EVERY_original_slot · Uses: G001, S009 S011 · impliesBefore source roles = [5, 3, 0, 2, 1, 1] After derived = [1] Conditions: none Apply the SAME codomain-label transposition to EVERY original source output,without reordering sources or changing only the first occurrence.Equal a,b is a legal no-op. Step identifiers and dependenciesRule: Bijection_OUTPUT_label_transposition_EVERY_original_slot · Uses: G001, S010 S012 · impliesBefore source roles = [5, 3, 0, 2, 2, 1] After derived = [1] Conditions: none Apply the SAME codomain-label transposition to EVERY original source output,without reordering sources or changing only the first occurrence.Equal a,b is a legal no-op. Step identifiers and dependenciesRule: Bijection_OUTPUT_label_transposition_EVERY_original_slot · Uses: G001, S011 S013 · impliesBefore source roles = [5, 3, 0, 2, 3, 0] After derived = [2] Conditions: none Apply the SAME codomain-label transposition to EVERY original source output,without reordering sources or changing only the first occurrence.Equal a,b is a legal no-op. Step identifiers and dependenciesRule: Bijection_OUTPUT_label_transposition_EVERY_original_slot · Uses: G001, S012 S014 · impliesBefore source roles = [5, 3, 0, 2, 4, 0] After derived = [2] Conditions: none Apply the SAME codomain-label transposition to EVERY original source output,without reordering sources or changing only the first occurrence.Equal a,b is a legal no-op. Step identifiers and dependenciesRule: Bijection_OUTPUT_label_transposition_EVERY_original_slot · Uses: G001, S013 S015 · calculationBefore Bijection TRANSPOSED reverse ALL fiber sizes through target0 := sum(0, 0) After Bijection TRANSPOSED reverse ALL fiber sizes through target0 = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore source roles = [Bijection TRANSPOSED, [1, 1, 1, 2, 2], 3, 0] After derived = [[], 0] Conditions: none For EVERY declared target list ALL ORIGINAL source preimages,including empty fibers,not merely attained values.Retain original labels/index order. Step identifiers and dependenciesRule: Bijection_EVERY_declared_target_COMPLETE_fiber_including_EMPTY · Uses: G001, S015 S017 · calculationBefore Bijection TRANSPOSED reverse ALL fiber sizes through target1 := sum(0, 3) After Bijection TRANSPOSED reverse ALL fiber sizes through target1 = 3 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore source roles = [Bijection TRANSPOSED, [1, 1, 1, 2, 2], 3, 1] After derived = [[0, 1, 2], 3] Conditions: none For EVERY declared target list ALL ORIGINAL source preimages,including empty fibers,not merely attained values.Retain original labels/index order. Step identifiers and dependenciesRule: Bijection_EVERY_declared_target_COMPLETE_fiber_including_EMPTY · Uses: G001, S017 S019 · calculationBefore Bijection TRANSPOSED reverse ALL fiber sizes through target2 := sum(3, 2) After Bijection TRANSPOSED reverse ALL fiber sizes through target2 = 5 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore source roles = [Bijection TRANSPOSED, [1, 1, 1, 2, 2], 3, 2] After derived = [[3, 4], 2] Conditions: none For EVERY declared target list ALL ORIGINAL source preimages,including empty fibers,not merely attained values.Retain original labels/index order. Step identifiers and dependenciesRule: Bijection_EVERY_declared_target_COMPLETE_fiber_including_EMPTY · Uses: G001, S019 S021 · impliesBefore source roles = [Bijection TRANSPOSED, [1, 1, 1, 2, 2], 3] After derived = [[[], [0, 1, 2], [3, 4]], [[0, 1], [0, 2], [1, 2], [3, 4]], [0], No, No, No, 5] Conditions: none A collision has xempty has all three;empty->nonempty only injectivity. Step identifiers and dependenciesRule: Bijection_ALL_collisions_GLOBAL_LEX_missing_targets_THREE_distinct_properties · Uses: G001, S020 S022 · impliesBefore source roles = [5, 3, [1, 1, 1, 0, 0], 0, 2] After derived = [[1, 1, 1, 2, 2], [[], [0, 1, 2], [3, 4]], No, No, Yes, [0, 1, 2]] Conditions: none A TYPED codomain transposition restores EVERY label when applied twice,so it preserves bijectivity eligibility.The original map need not be bijective.False/False still has agreementTrue;do not confuse preservation with original bijectivity.At m0 no valid a,b exist for THIS family. Step identifiers and dependenciesRule: Bijection_transposition_is_codomain_PERMUTATION_flags_AGREEMENT · Uses: G001, S021 ## Reverse checks - S022 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S021 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S020 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S019 · premise_and_result_audit — Recompute Bijection TRANSPOSED reverse ALL fiber sizes through target2 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S017 · premise_and_result_audit — Recompute Bijection TRANSPOSED reverse ALL fiber sizes through target1 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S015 · premise_and_result_audit — Recompute Bijection TRANSPOSED reverse ALL fiber sizes through target0 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S013 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S012 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S011 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S010 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S009 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S008 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S007 · premise_and_result_audit — Recompute Bijection ORIGINAL reverse ALL fiber sizes through target2 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S005 · premise_and_result_audit — Recompute Bijection ORIGINAL reverse ALL fiber sizes through target1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S003 · premise_and_result_audit — Recompute Bijection ORIGINAL reverse ALL fiber sizes through target0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S001 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. ## Result ([1, 1, 1, 0, 0], [1, 1, 1, 2, 2], [[], [0, 1, 2], [3, 4]], No, No, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 5e6d843d7feb828f019a19c9b1c438105312dd12ad4f4731e9315dbc074fe87d - candidate record SHA-256: 38b2a5ec74c357426da8f92cbe4282da08acf5e4639a79efc86699ab7b7a040a - semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf - question-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r01-bijective-0067.json)