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Question 5080: Complete Finite Powerset

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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← Question derivation index · Original question batch

R0009-05080 · q-20261008-advance-20261011-r04-set-0080

Original question and givens

Finite integer set A=[-4, 0, 1, 4]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

  • G001 — Finite integer set A=[-4, 0, 1, 4]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

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

raw integer list=[-4, 0, 1, 4]

After

extensional set=[-4, 0, 1, 4]

Conditions: none

A set depends on its elements;deduplication and numeric ordering do not change membership.The original list is retained as source.

Step identifiers and dependencies

Rule: set_extensionality_normalization · Uses: G001

S002 · implies

Before

no elements processed

After

[[]]

Conditions: none

The empty set has one subset,the empty set itself;its power set is not an empty list.

Step identifiers and dependencies

Rule: power_set_empty_initial · Uses: G001, S001

S003 · implies

Before

processed subset=[];next element=-4

After

omit=[];include=[-4]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S002

S004 · implies

Before

processed subset=[];next element=0

After

omit=[];include=[0]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S003

S005 · implies

Before

processed subset=[-4];next element=0

After

omit=[-4];include=[-4, 0]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S004

S006 · implies

Before

processed subset=[];next element=1

After

omit=[];include=[1]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S005

S007 · implies

Before

processed subset=[-4];next element=1

After

omit=[-4];include=[-4, 1]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S006

S008 · implies

Before

processed subset=[0];next element=1

After

omit=[0];include=[0, 1]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S007

S009 · implies

Before

processed subset=[-4, 0];next element=1

After

omit=[-4, 0];include=[-4, 0, 1]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S008

S010 · implies

Before

processed subset=[];next element=4

After

omit=[];include=[4]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S009

S011 · implies

Before

processed subset=[-4];next element=4

After

omit=[-4];include=[-4, 4]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S010

S012 · implies

Before

processed subset=[0];next element=4

After

omit=[0];include=[0, 4]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S011

S013 · implies

Before

processed subset=[-4, 0];next element=4

After

omit=[-4, 0];include=[-4, 0, 4]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S012

S014 · implies

Before

processed subset=[1];next element=4

After

omit=[1];include=[1, 4]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S013

S015 · implies

Before

processed subset=[-4, 1];next element=4

After

omit=[-4, 1];include=[-4, 1, 4]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S014

S016 · implies

Before

processed subset=[0, 1];next element=4

After

omit=[0, 1];include=[0, 1, 4]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S015

S017 · implies

Before

processed subset=[-4, 0, 1];next element=4

After

omit=[-4, 0, 1];include=[-4, 0, 1, 4]

Conditions: none

Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.

Step identifiers and dependencies

Rule: power_set_include_omit_branch · Uses: G001, S016

S018 · implies

Before

(2)^4

After

power-set cardinality = 16

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependencies

Rule: integer_power · Uses: G001, S017

S019 · implies

Before

[[], [-4], [0], [1], [4], [-4, 0], [-4, 1], [-4, 4], [0, 1], [0, 4], [1, 4], [-4, 0, 1], [-4, 0, 4], [-4, 1, 4], [0, 1, 4], [-4, 0, 1, 4]]

After

all 16 subsets in size-then-numeric-lex order

Conditions: none

Order subsets first by size,then numeric lexicographic order,with increasing elements inside each subset.This constructs only subsets of the supplied finite A,not unrestricted comprehension.

Step identifiers and dependencies

Rule: finite_power_set_completeness · Uses: G001, S018

Reverse checks

  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

[[], [-4], [0], [1], [4], [-4, 0], [-4, 1], [-4, 4], [0, 1], [0, 4], [1, 4], [-4, 0, 1], [-4, 0, 4], [-4, 1, 4], [0, 1, 4], [-4, 0, 1, 4]]

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: 10f98f72defe3b7fedee0e0ef27813adbc92e15a796fcf8e418d657228992ccc
  • candidate record SHA-256: b62e61223d7c82e046f860bcdf7e84c268472967c500b755e539f9843439f0db
  • semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
  • question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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