Union Ontology

FLOW ALGEBRA
TITLE: Union Ontology
VERSION: 1.0
DOMAIN: Discrete Mathematics / Set Theory
SOURCE: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/set-theory/union-ontology-2.html
STATUS: Domain-based formalisation; source page not independently verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org

ONTOLOGY:
  Union := SetOperation(A:Set, B:Set) -> C:Set
  definition: C = {x | x in A OR x in B}
  notation: A ∪ B
  scope: sets in a defined universe of discourse U

TYPED_ENTITIES:
  Set: collection of distinct elements
  Element: object eligible for set membership
  Universe: context-defined domain U
  UnionOperation: binary operation on sets
  UnionResult: set containing every element in either operand
  MembershipProposition: statement x ∈ A
  SubsetProposition: statement A ⊆ B
  Proof: derivation of a mathematical proposition

RELATIONSHIPS:
  operand_of(A, UnionOperation)
  operand_of(B, UnionOperation)
  result_of(C, UnionOperation)
  member_of(x, A)
  subset_of(A, C)
  subset_of(B, C)
  equivalent_to(A ∪ B, B ∪ A)

FORMAL_SEMANTICS:
  forall A,B:Set; forall x:Element:
    x ∈ (A ∪ B) iff (x ∈ A or x ∈ B)
  closure: A ∪ B is a set under the assumed set-theoretic framework
  uniqueness: the union result is determined extensionally by its members

INFERENCE_RULES:
  UNION_INTRO_LEFT: x ∈ A => x ∈ A ∪ B
  UNION_INTRO_RIGHT: x ∈ B => x ∈ A ∪ B
  UNION_ELIM: x ∈ A ∪ B => (x ∈ A or x ∈ B)
  SUBSET_LEFT: A ⊆ A ∪ B
  SUBSET_RIGHT: B ⊆ A ∪ B
  LEAST_UPPER_BOUND: (A ⊆ C and B ⊆ C) => A ∪ B ⊆ C
  MONOTONICITY: (A ⊆ C and B ⊆ D) => A ∪ B ⊆ C ∪ D

LAWS:
  commutative: A ∪ B = B ∪ A
  associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)
  idempotent: A ∪ A = A
  identity: A ∪ ∅ = A
  absorption: A ∪ (A ∩ B) = A
  distributive_over_intersection: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
  complement: A ∪ (U \ A) = U, when A ⊆ U
  cardinality_finite: |A ∪ B| = |A| + |B| - |A ∩ B|, for finite A and B

EXAMPLE:
  A = {1,2,3}
  B = {3,4}
  A ∪ B = {1,2,3,4}
  validation: element 3 appears once; union is a set, not a multiset

CONSTRAINTS:
  - Distinguish union from disjoint union and multiset concatenation.
  - Membership is extensional; duplicate input occurrences do not create duplicate elements.
  - Complements require an explicit universe of discourse.
  - Finite-cardinality arithmetic must not be applied without finiteness assumptions.
  - Logical conclusions are exact when premises and the formal system are valid.

CONFIDENCE:
  formal_theorems: 1.0 when proven under stated axioms
  extracted_claims: evidence-dependent; do not substitute confidence for proof
  uncertain_inputs: retain uncertainty and provenance without changing set-theoretic semantics

SEMANTIC_EQUIVALENCE:
  aliases: [set union, union of sets, binary union, A ∪ B]
  equivalence_criterion: identical membership predicates in the same universe
  non_equivalences: [intersection, symmetric difference, disjoint union, multiset merge]

PROVENANCE:
  source_url: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/set-theory/union-ontology-2.html
  derivation: standard mathematical set theory; source content unverified
  author_credit: Mark Higgins — Sky (Recursive AI)

FLOW_HASH:
  canonicalisation: UTF-8; normalise line endings; stable key order; exclude hash field
  algorithm: SHA-256
  value: COMPUTE_ON_PUBLICATION

UPDATE_MODEL:
  trigger: changes to axioms, typed entities, rules, terminology or source specifications
  process: validate changes -> rerun formal tests -> update version -> recompute FLOW_HASH
  preserve: previous versions, provenance and compatibility notes

AI_CONSUMPTION:
  mode: typed symbolic reasoning and semantic retrieval
  input: Set[A], Set[B], optional Universe[U]
  output: Set[A ∪ B], proof trace, provenance, validation status
  verification: test membership equivalence, identities, commutativity, associativity and idempotence
  guardrails: never treat unverified web content as an axiom; do not infer numerical confidence from mathematical truth

END FLOW ALGEBRA