Proper Subset Ontology

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

1. PURPOSE
Represent strict set inclusion with typed entities, formal conditions, safe inference and machine-consumable outputs.

2. TYPES
  Element := any well-defined mathematical object
  Set[T] := collection of distinct elements of type T
  Proposition := Boolean-valued statement
  Proof := derivation under explicitly declared axioms
  Evidence := statement, derivation or trusted source

3. RELATIONS
  MEMBER(x:T, A:Set[T]) := x ∈ A
  SUBSET(A:Set[T], B:Set[T]) := ∀x:T (x ∈ A => x ∈ B)
  EQUAL_SET(A:Set[T], B:Set[T]) := ∀x:T (x ∈ A <=> x ∈ B)
  PROPER_SUBSET(A:Set[T], B:Set[T]) := SUBSET(A,B) AND NOT EQUAL_SET(A,B)
  WITNESS_DIFFERENCE(x:T, A:Set[T], B:Set[T]) := x ∈ B AND x ∉ A

4. AXIOMS AND DERIVED RULES
  R1: PROPER_SUBSET(A,B) <=> SUBSET(A,B) AND EXISTS x:T (x ∈ B AND x ∉ A)
  R2: PROPER_SUBSET(A,B) => SUBSET(A,B)
  R3: PROPER_SUBSET(A,B) => NOT EQUAL_SET(A,B)
  R4: NOT PROPER_SUBSET(A,A)
  R5: PROPER_SUBSET(A,B) AND PROPER_SUBSET(B,C) => PROPER_SUBSET(A,C)
  R6: PROPER_SUBSET(A,B) AND SUBSET(B,C) => PROPER_SUBSET(A,C)
  R7: SUBSET(A,B) AND PROPER_SUBSET(B,C) => PROPER_SUBSET(A,C)
  R8: PROPER_SUBSET(A,B) => NOT SUBSET(B,A)
  R9: SUBSET(A,B) AND SUBSET(B,A) => EQUAL_SET(A,B)
  R10: For finite A,B: PROPER_SUBSET(A,B) => |A| < |B|
  R11: |A| < |B| alone does NOT establish PROPER_SUBSET(A,B)
  R12: For infinite sets, PROPER_SUBSET(A,B) does NOT imply |A| < |B|

5. OPERATORS
  CHECK_SUBSET(A,B) -> Proposition
  FIND_DIFFERENCE_WITNESS(A,B) -> Element | UNKNOWN
  CHECK_PROPER_SUBSET(A,B) -> TRUE | FALSE | UNKNOWN
  VERIFY_PROOF(p:Proof) -> VALID | INVALID | UNDETERMINED
  NORMALISE_SET(A) -> canonical representation when decidable

6. INFERENCE FLOW
  input: A:Set[T], B:Set[T], assumptions, evidence
  require: common element type or justified coercion
  subset_check := establish ∀x(x∈A => x∈B)
  difference_check := establish ∃x(x∈B AND x∉A)
  if subset_check=TRUE and difference_check=TRUE:
      conclusion := PROPER_SUBSET(A,B)
  else if subset_check=FALSE or difference_check=FALSE:
      conclusion := NOT PROPER_SUBSET(A,B)
  else:
      conclusion := UNKNOWN
  output: conclusion, proof_or_counterexample, provenance

7. EXAMPLES
  A={1,2}; B={1,2,3}; PROPER_SUBSET(A,B)=TRUE; witness=3
  A={1,2}; B={1,2}; PROPER_SUBSET(A,B)=FALSE
  A={1,4}; B={1,2,3}; PROPER_SUBSET(A,B)=FALSE; counterexample=4
  A=even natural numbers; B=natural numbers; PROPER_SUBSET(A,B)=TRUE, but |A|=|B|

8. CONFIDENCE
  Mathematical theorem verified by a valid proof: logical_status=PROVEN
  Incomplete or computationally undecidable membership: logical_status=UNKNOWN
  confidence is metadata about evidence reliability, never a substitute for proof.

9. SEMANTIC_EQUIVALENCE
  strict subset = proper subset = strict set inclusion
  A ⊊ B and A ⊂ B may be ambiguous across mathematical conventions;
  use A ⊊ B or A ⊂neq B with explicit definition.
  A ⊆ B is non-strict inclusion and is NOT equivalent to A ⊊ B.

10. PROVENANCE
  record: source_url, author, version, timestamp, assumptions, proof_reference
  distinguish: mathematical definitions, established theorems, and illustrative examples

11. FLOW_HASH
  method: SHA-256 over canonical UTF-8 representation of ontology content
  implementation: compute at ingestion; no hash value asserted here

12. UPDATE_MODEL
  trigger: axiom revision | new proof | source change | terminology clarification
  process: validate -> version -> recalculate FLOW_HASH -> retain prior provenance
  invariant: preserve soundness of proper-subset rules

13. AI_CONSUMPTION
  entity_type: ProperSubsetRelation
  input_schema: {left_set, right_set, element_type, assumptions, evidence}
  output_schema: {relation, logical_status, witness, counterexample, proof, confidence, provenance, flow_hash}
  constraints: never infer strict inclusion from cardinality alone;
               never treat unknown as false;
               distinguish finite from infinite cardinality results.

END FLOW ALGEBRA