OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 4495: Union and Intersection Cardinalities

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

Return to catalogue page 23

← Question derivation index · Original question batch

R0007-04495 · q-20261008-advance-20261010-r02-set-operations-0020

Original question and givens

Finite integer sets A=[0, -1] and B=[0, 100, 101]; repeated elements count once. Give (|A∪B|, |A∩B|).

  • G001 — Finite integer sets A=[0, -1] and B=[0, 100, 101]; repeated elements count once. Give (|A∪B|, |A∩B|).

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 A=[0, -1];raw B=[0, 100, 101]

After

A=[-1, 0];B=[0, 100, 101]

Conditions: none

Deduplicate and numerically sort integer sets;repetition and input order do not change membership.The original lists remain retained.

Step identifiers and dependencies

Rule: finite_set_extensionality · Uses: G001

S002 · implies

Before

element=-1

After

in A=Yes;in B=No

Conditions: none

Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting.

Step identifiers and dependencies

Rule: finite_set_membership_row · Uses: G001, S001

S003 · implies

Before

element=0

After

in A=Yes;in B=Yes

Conditions: none

Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting.

Step identifiers and dependencies

Rule: finite_set_membership_row · Uses: G001, S002

S004 · implies

Before

element=100

After

in A=No;in B=Yes

Conditions: none

Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting.

Step identifiers and dependencies

Rule: finite_set_membership_row · Uses: G001, S003

S005 · implies

Before

element=101

After

in A=No;in B=Yes

Conditions: none

Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting.

Step identifiers and dependencies

Rule: finite_set_membership_row · Uses: G001, S004

S006 · calculation

Before

separate set cardinalities sum := sum(2, 3)

After

separate set cardinalities sum = 5

Conditions: none

Adding the two set sizes counts their intersection twice.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

union cardinality check := sub(5, 1)

After

union cardinality check = 4

Conditions: none

Subtract one intersection count and compare with the deduplicated union size.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

union=[-1, 0, 100, 101];intersection=[0]

After

(4, 1)

Conditions: none

Report union size before intersection size;the empty set has size0 and a shared element is counted once in the union.

Step identifiers and dependencies

Rule: union_intersection_cardinality · Uses: G001, S007

Reverse checks

  • 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 — Recompute union cardinality check with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute separate set cardinalities sum with exact rational arithmetic, not rounded decimals.
  • 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, 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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 3dd093b9d1c80dc22b4be117fc114974606aa78f7c7f58f2b37939e4113d7071
  • candidate record SHA-256: 0ae3902232ed4dbb50b192f30a292363fc3390210b002fda272c57d757e8b367
  • semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a
  • question-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON