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Question 32442: Cardinality of a Specified Fiber

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0063-32442 · q-20261004-advance-20261005-r01-injective-0067

Original question and givens

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. For target t=-2∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

  • G001 — The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. For target t=-2∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-injective-fiber-cardinality, fiber_cardinality, no-required-domain, []]

After

derived = [[integer_fiber_cardinality_STRING]]

Conditions: none

BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds.

Step identifiers and dependencies

Rule: R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle · Uses: G001

S002 · implies

Before

source roles = [A declared domain, legacy-injective, [-4, -3, -2, -1, 0]]

After

derived = [[-4, -3, -2, -1, 0]]

Conditions: none

RAW A/B/C are distinct SAFE integers.Legacy injective limits100,but PULSE has NO such cap.Sort ONLY a derived work table,retaining original lists.

Step identifiers and dependencies

Rule: Map_RAW_distinct_SAFE_set_BANK_specific_cardinality_bound · Uses: G001, S001

S003 · implies

Before

source roles = [B declared codomain, legacy-injective, [-6, -5, -4, -3, -2, -1]]

After

derived = [[-6, -5, -4, -3, -2, -1]]

Conditions: none

RAW A/B/C are distinct SAFE integers.Legacy injective limits100,but PULSE has NO such cap.Sort ONLY a derived work table,retaining original lists.

Step identifiers and dependencies

Rule: Map_RAW_distinct_SAFE_set_BANK_specific_cardinality_bound · Uses: G001, S002

S004 · implies

Before

source roles = [legacy-injective, [[-4, -1], [-3, -1], [-2, -2], [-1, -2], [0, -3]]]

After

derived = [[[-4, -1], [-3, -1], [-2, -2], [-1, -2], [0, -3]]]

Conditions: none

PULSE rejects EXACT duplicate RAW rows BEFORE totality,not as a legal No.Legacy injective FIRST checks raw length≤200,then normalizes identical rows as one pair.Do not exchange these policies;type-check EVERY raw cell first.

Step identifiers and dependencies

Rule: Map_RAW_pairs_arity_SAFE_DUPLICATE_policy_BEFORE_totality · Uses: G001, S003

S005 · implies

Before

source roles = [legacy-injective, 0, -4, -1, [-4, -3, -2, -1, 0], [-6, -5, -4, -3, -2, -1]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S004

S006 · implies

Before

source roles = [legacy-injective, 1, -3, -1, [-4, -3, -2, -1, 0], [-6, -5, -4, -3, -2, -1]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S005

S007 · implies

Before

source roles = [legacy-injective, 2, -2, -2, [-4, -3, -2, -1, 0], [-6, -5, -4, -3, -2, -1]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S006

S008 · implies

Before

source roles = [legacy-injective, 3, -1, -2, [-4, -3, -2, -1, 0], [-6, -5, -4, -3, -2, -1]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S007

S009 · implies

Before

source roles = [legacy-injective, 4, 0, -3, [-4, -3, -2, -1, 0], [-6, -5, -4, -3, -2, -1]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S008

S010 · implies

Before

source roles = [legacy-injective, -4, [[-4, -1], [-3, -1], [-2, -2], [-1, -2], [0, -3]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S009

S011 · implies

Before

source roles = [legacy-injective, -3, [[-4, -1], [-3, -1], [-2, -2], [-1, -2], [0, -3]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S010

S012 · implies

Before

source roles = [legacy-injective, -2, [[-4, -1], [-3, -1], [-2, -2], [-1, -2], [0, -3]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S011

S013 · implies

Before

source roles = [legacy-injective, -1, [[-4, -1], [-3, -1], [-2, -2], [-1, -2], [0, -3]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S012

S014 · implies

Before

source roles = [legacy-injective, 0, [[-4, -1], [-3, -1], [-2, -2], [-1, -2], [0, -3]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S013

S015 · implies

Before

source roles = [legacy-injective, [-4, -3, -2, -1, 0], [-6, -5, -4, -3, -2, -1], [[-4, -1], [-3, -1], [-2, -2], [-1, -2], [0, -3]], Yes]

After

derived = [[Yes, Yes, Yes, Yes, Yes], [1, 1, 1, 1, 1], Yes]

Conditions: none

A total function needs BOTH every declared input AND every row qualified.PULSE validity may legally return No;other queries and legacy injective REQUIRE actual totality BEFORE lookup.Empty A/table can be valid;unattained B labels remain.

Step identifiers and dependencies

Rule: Map_TOTAL_all_declared_inputs_AND_ALL_rows_before_any_lookup · Uses: G001, S014

S016 · implies

Before

source roles = [-4, -1, -2, [-6, -5, -4, -3, -2, -1]]

After

derived = [No]

Conditions: none

t is a target IN B,not an input.Enumerate ALL A preimages with output t,not just the first and not weights/probabilities.

Step identifiers and dependencies

Rule: Injective_FIBER_EACH_input_compare_to_TYPED_codomain_target · Uses: G001, S015

S017 · implies

Before

source roles = [-3, -1, -2, [-6, -5, -4, -3, -2, -1]]

After

derived = [No]

Conditions: none

t is a target IN B,not an input.Enumerate ALL A preimages with output t,not just the first and not weights/probabilities.

Step identifiers and dependencies

Rule: Injective_FIBER_EACH_input_compare_to_TYPED_codomain_target · Uses: G001, S016

S018 · implies

Before

source roles = [-2, -2, -2, [-6, -5, -4, -3, -2, -1]]

After

derived = [Yes]

Conditions: none

t is a target IN B,not an input.Enumerate ALL A preimages with output t,not just the first and not weights/probabilities.

Step identifiers and dependencies

Rule: Injective_FIBER_EACH_input_compare_to_TYPED_codomain_target · Uses: G001, S017

S019 · calculation

Before

Injective fiber ORIGINAL preimage -2 count := sum(0, 1)

After

Injective fiber ORIGINAL preimage -2 count = 1

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

source roles = [-1, -2, -2, [-6, -5, -4, -3, -2, -1]]

After

derived = [Yes]

Conditions: none

t is a target IN B,not an input.Enumerate ALL A preimages with output t,not just the first and not weights/probabilities.

Step identifiers and dependencies

Rule: Injective_FIBER_EACH_input_compare_to_TYPED_codomain_target · Uses: G001, S019

S021 · calculation

Before

Injective fiber ORIGINAL preimage -1 count := sum(1, 1)

After

Injective fiber ORIGINAL preimage -1 count = 2

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

source roles = [0, -3, -2, [-6, -5, -4, -3, -2, -1]]

After

derived = [No]

Conditions: none

t is a target IN B,not an input.Enumerate ALL A preimages with output t,not just the first and not weights/probabilities.

Step identifiers and dependencies

Rule: Injective_FIBER_EACH_input_compare_to_TYPED_codomain_target · Uses: G001, S021

S023 · implies

Before

source roles = [[-4, -3, -2, -1, 0], -2, [-2, -1]]

After

derived = [2]

Conditions: none

The COMPLETE fiber may be empty with cardinality0.Return its count as an integer string,not the target value or a probability.

Step identifiers and dependencies

Rule: Injective_FIBER_complete_zero_size_legal_cardinality_STRING · Uses: G001, S022

Reverse checks

  • S023 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S022 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S021 · premise_and_result_audit — Recompute Injective fiber ORIGINAL preimage -1 count with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S019 · premise_and_result_audit — Recompute Injective fiber ORIGINAL preimage -2 count with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S017 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S016 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S015 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S014 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S013 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S012 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S011 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S010 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S009 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S008 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S007 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S006 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S005 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S004 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S003 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S002 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S001 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: df840bb2422f9a471a8ea5412f16cb76cbf1fbdc292c30d2ccd5084839009f67
  • candidate record SHA-256: be2cd67d1678f879000e5594fe1264651921a1c8a142f2da568a341dad517f3f
  • semantic review receipt SHA-256: acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0
  • question-first witness SHA-256: eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON