Foundations of Discrete Mathematics

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
Title: Foundations of Discrete Mathematics
Version: 1.0
Source: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/foundations-of-discrete-mathematics.html
Authors: Mark Anthony Higgins — Sky (Recursive AI)
Source scope: Main-page section index; formal domain-level synthesis, not an extraction of child-page specifications.

AI_INTERPRETATION_NOTE:
  Interpret each typed assertion and inference as a formal structural rule.
  Preserve mathematical definitions and distinguish proven results from conjectures.

DOMAIN:
  id: foundations_of_discrete_mathematics
  parent: discrete_mathematics_ontologies
  kind: mathematical_foundations

CHILD_ONTOLOGIES:
  - discrete_mathematics_ontology
  - finite_mathematics_ontology
  - discrete_structures_ontology
  - countability_ontology
  - mathematical_objects_ontology
  - finite_set_ontology
  - infinite_set_ontology
  - mathematical_relation_ontology
  - mathematical_structure_ontology
  - discrete_model_ontology

TYPES:
  MathematicalObject: abstract_entity
  Set: MathematicalObject
  FiniteSet: Set
  InfiniteSet: Set
  Element: MathematicalObject
  Relation: MathematicalObject
  Function: Relation
  Structure: MathematicalObject
  DiscreteStructure: Structure
  DiscreteModel: MathematicalObject
  Cardinality: MathematicalObject
  NaturalNumber: MathematicalObject
  Proposition: MathematicalObject
  Proof: MathematicalObject
  Axiom: Proposition

RELATIONSHIPS:
  member_of(Element, Set) -> Boolean
  subset_of(Set, Set) -> Boolean
  related_by(Element, Element, Relation) -> Boolean
  has_domain(Relation, Set) -> Boolean
  has_codomain(Relation, Set) -> Boolean
  has_cardinality(Set, Cardinality) -> Boolean
  models(DiscreteModel, Structure) -> Boolean
  satisfies(DiscreteModel, Proposition) -> Boolean
  proves(Proof, Proposition) -> Boolean
  depends_on(Proposition, Proposition) -> Boolean
  equivalent_to(MathematicalObject, MathematicalObject) -> Boolean

AXIOMS_AND_RULES:
  R01: finite(S:Set) iff exists n:NaturalNumber and bijection(S, {0,...,n-1}).
  R02: infinite(S:Set) iff not finite(S).
  R03: countably_infinite(S:Set) iff exists bijection(S, NaturalNumbers).
  R04: countable(S:Set) iff finite(S) or countably_infinite(S).
  R05: subset_of(A,B) iff forall x (member_of(x,A) implies member_of(x,B)).
  R06: equivalence_relation(R,S) iff reflexive(R,S) and symmetric(R,S) and transitive(R,S).
  R07: partial_order(R,S) iff reflexive(R,S) and antisymmetric(R,S) and transitive(R,S).
  R08: function(f,A,B) iff forall a in A exists exactly_one b in B with related_by(a,b,f).
  R09: isomorphic(A:Structure,B:Structure) implies structural_equivalence(A,B), relative to the declared signature.
  R10: satisfies(M,P) and valid_inference(P,Q) implies satisfies(M,Q).
  R11: no cardinality, isomorphism, or theorem claim may be inferred without applicable definitions and evidence.

OPERATORS:
  DEFINE(entity, type, conditions) -> TypedEntity
  CLASSIFY(set) -> {finite, countably_infinite, uncountable, unknown}
  CONSTRUCT_RELATION(domain, codomain, pairs) -> Relation
  VERIFY_AXIOMS(structure, axioms) -> VerificationResult
  INFER(premises, valid_rule) -> Proposition
  VALIDATE(proposition, proof_or_model) -> ValidationResult
  TRACE(result) -> ProvenanceRecord

CONFIDENCE:
  type: real in [0,1]
  semantics: evidence_assessment_not_mathematical_truth
  formal_proof: requires independently checked proof and explicit assumptions
  uncertain_claim: retain uncertainty; never promote to theorem automatically

SEMANTIC_EQUIVALENCE:
  require: matching definitions, domains, signatures, and logical conditions
  aliases:
    finite_cardinality: finite_set_size
    denumerable: countably_infinite
  warning: countable != finite; equinumerous != identical

PROVENANCE:
  source_url: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/foundations-of-discrete-mathematics.html
  source_page_verified: true
  child_page_contents_verified: false
  revision_policy: record source revision, author, assumptions and validation status

FLOW_HASH:
  method: SHA-256
  input: canonical_serialisation(DOMAIN, CHILD_ONTOLOGIES, TYPES, RELATIONSHIPS, AXIOMS_AND_RULES, OPERATORS)
  note: hash calculated by consuming implementation; no uncomputed hash asserted

UPDATE_MODEL:
  on_source_change:
    - compare section names and links
    - reconcile entity types and mappings
    - rerun formal validation and dependency checks
    - recalculate FLOW_HASH
    - retain version history and provenance

AI_CONSUMPTION:
  parse_as: typed_formal_knowledge
  reasoning: explicit_rule_application
  output: {conclusion, assumptions, proof_or_evidence, confidence, provenance}
  safeguards:
    - distinguish syntax from semantics
    - do not conflate finite, countable, and uncountable
    - do not treat probabilistic confidence as proof
    - do not infer child-page details from section titles

END FLOW ALGEBRA