Complement Ontology

KIRETH RESEARCH — SEMANTIC LIBRARY
AI CORE → ONTOLOGIES → MATHEMATICAL ONTOLOGIES → MATHEMATICAL LOGIC ONTOLOGIES
SET THEORY → COMPLEMENT ONTOLOGY

Canonical URL:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/set-theory/complement-ontology.html

FLOW_ALGEBRA ComplementOntology_v1.0

BEGIN_FLOW

DEFINE DOMAIN ComplementOntology := {
    Complement,
    Set,
    Element,
    Membership,
    NonMembership,
    SetTheory,
    UniversalSet,
    RelativeUniverse,
    RelativeComplement,
    SetDifference,
    EmptySet,
    Subset,
    Union,
    Intersection,
    Difference,
    SymmetricDifference,
    PowerSet,
    DeMorganLaw,
    Involution,
    BooleanAlgebra,
    Context,
    DomainOfDiscourse
}

DEFINE ENTITY_TYPES := {
    SET_OPERATION,
    SET,
    ELEMENT,
    UNIVERSAL_SET,
    CONTEXT,
    SET_RELATION,
    ALGEBRAIC_PROPERTY,
    SEMANTIC_OBJECT
}

DEFINE RELATION_TYPES := {
    MEMBER_OF,
    NOT_MEMBER_OF,
    SUBSET_OF,
    COMPLEMENT_OF,
    RELATIVE_TO,
    CONTAINS,
    EQUAL_TO,
    DISJOINT_WITH,
    DEPENDS_ON,
    IS_INSTANCE_OF,
    IS_SUBTYPE_OF
}

SET Complement.type = SET_OPERATION
SET Complement.context = SetTheory
SET Complement.role =
    "construct, relative to a specified universe, the set of elements that belong to the universe but do not belong to the target set"

RULE complement_definition:
    UniversalSet(U)
    ∧ Set(A)
    ∧ SUBSET_OF(A,U)
    -> Complement_U(A)
       = U \ A

RULE complement_membership:
    UniversalSet(U)
    ∧ SUBSET_OF(A,U)
    -> x ∈ Complement_U(A)
       IFF
       (x ∈ U AND x ∉ A)

RULE complement_nonmembership:
    UniversalSet(U)
    ∧ SUBSET_OF(A,U)
    ∧ x ∈ A
    -> x ∉ Complement_U(A)

RULE complement_subset_universe:
    UniversalSet(U)
    ∧ SUBSET_OF(A,U)
    -> SUBSET_OF(Complement_U(A),U)

RULE complement_disjointness:
    UniversalSet(U)
    ∧ SUBSET_OF(A,U)
    -> A ∩ Complement_U(A) = ∅
    ∧ DISJOINT_WITH(A,Complement_U(A))

RULE complement_exhaustion:
    UniversalSet(U)
    ∧ SUBSET_OF(A,U)
    -> A ∪ Complement_U(A) = U

RULE complement_involution:
    UniversalSet(U)
    ∧ SUBSET_OF(A,U)
    -> Complement_U(Complement_U(A)) = A

RULE complement_empty_set:
    UniversalSet(U)
    -> Complement_U(∅) = U

RULE complement_universal_set:
    UniversalSet(U)
    -> Complement_U(U) = ∅

RULE complement_subset_reversal:
    UniversalSet(U)
    ∧ SUBSET_OF(A,B)
    ∧ SUBSET_OF(B,U)
    -> SUBSET_OF(Complement_U(B),Complement_U(A))

RULE DeMorgan_union:
    UniversalSet(U)
    ∧ SUBSET_OF(A,U)
    ∧ SUBSET_OF(B,U)
    -> Complement_U(A ∪ B)
       = Complement_U(A) ∩ Complement_U(B)

RULE DeMorgan_intersection:
    UniversalSet(U)
    ∧ SUBSET_OF(A,U)
    ∧ SUBSET_OF(B,U)
    -> Complement_U(A ∩ B)
       = Complement_U(A) ∪ Complement_U(B)

RULE relative_complement:
    Set(A)
    ∧ Set(B)
    -> RelativeComplement(A,B)
       = A \ B
       = {x | x ∈ A AND x ∉ B}

RULE complement_as_relative_difference:
    UniversalSet(U)
    ∧ SUBSET_OF(A,U)
    -> Complement_U(A)
       = RelativeComplement(U,A)

RULE difference_via_complement:
    UniversalSet(U)
    ∧ SUBSET_OF(A,U)
    ∧ SUBSET_OF(B,U)
    -> A \ B
       = A ∩ Complement_U(B)

RULE symmetric_difference_via_complements:
    UniversalSet(U)
    ∧ SUBSET_OF(A,U)
    ∧ SUBSET_OF(B,U)
    -> A △ B
       = (A ∩ Complement_U(B))
         ∪
         (B ∩ Complement_U(A))

RULE complement_context_dependence:
    Set(A)
    ∧ UniversalSet(U1)
    ∧ UniversalSet(U2)
    ∧ U1 != U2
    ∧ SUBSET_OF(A,U1)
    ∧ SUBSET_OF(A,U2)
    -> Complement_U1(A)
       MAY_DIFFER_FROM
       Complement_U2(A)

DEFINE COMPLEMENT_MODEL := {
    notation: {
        "Aᶜ",
        "A'",
        "U \ A"
    },
    required_context: UniversalSet,
    formal_definition:
        "Aᶜ_U = {x ∈ U | x ∉ A}",
    output_type: Set
}

DEFINE RELATIVE_COMPLEMENT_MODEL := {
    notation: "A \ B",
    formal_definition:
        "{x | x ∈ A and x ∉ B}",
    universe_requirement:
        "no separate universal set is required when A itself supplies the reference set"
}

DEFINE COMPLEMENT_IDENTITIES := {
    involution: "(Aᶜ)ᶜ = A",
    excluded_middle_set_form: "A ∪ Aᶜ = U",
    contradiction_set_form: "A ∩ Aᶜ = ∅",
    empty_complement: "∅ᶜ = U",
    universe_complement: "Uᶜ = ∅",
    subset_reversal: "A ⊆ B -> Bᶜ ⊆ Aᶜ"
}

DEFINE DE_MORGAN_MODEL := {
    union:
        "(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ",
    intersection:
        "(A ∩ B)ᶜ = Aᶜ ∪ Bᶜ",
    requirement:
        "all complements are taken relative to the same universe"
}

DEFINE EXAMPLE_1 := {
    universe: "U = {1,2,3,4,5}",
    A: "{1,3,5}",
    complement: "{2,4}"
}

DEFINE EXAMPLE_2 := {
    universe: "U = {a,b,c,d}",
    A: "{a,b}",
    identities: {
        "A ∪ Aᶜ = U",
        "A ∩ Aᶜ = ∅"
    }
}

DEFINE EXAMPLE_3 := {
    A: "{1,2,3,4}",
    B: "{3,4,5}",
    relative_complement: "A \ B = {1,2}"
}

DEFINE EXAMPLE_4 := {
    target_set: "{1}",
    universe_1: "{1,2,3}",
    universe_2: "{1,2,3,4}",
    complement_in_universe_1: "{2,3}",
    complement_in_universe_2: "{2,3,4}",
    observation: "set complement is universe-dependent"
}

DEFINE SEMANTIC_PIPELINE :=
    ComplementExpression
    -> IdentifyTargetSet
    -> ResolveReferenceUniverse
    -> ValidateTargetSubsetOfUniverse
    -> EnumerateOrCharacteriseUniverseElements
    -> ExcludeTargetMembers
    -> ConstructComplementSet
    -> ApplyComplementIdentities
    -> PreserveUniverseReference
    -> ReturnCanonicalComplement

DEFINE REASONING_PIPELINE :=
    InputSetExpression
    -> DetectComplementOrDifferenceNotation
    -> DistinguishAbsoluteNotationFromRelativeDifference
    -> ResolveUniversalSet
    -> ApplyMembershipNegationWithinUniverse
    -> ApplyInvolution
    -> ApplySubsetReversal
    -> ApplyDeMorganLaws
    -> SimplifyEmptyAndUniversalComplements
    -> ReturnDerivedRelations

DEFINE COMPLEMENT_GRAPH_MODEL := {
    universe_node: UniversalSet,
    target_node: Set,
    operation_node: Complement,
    result_node: Set,
    membership_rule:
        "x is in result iff x is in universe and not in target",
    required_edge:
        "target --SUBSET_OF--> universe",
    context_preservation: TRUE
}

DEFINE ONTOLOGY_RELATIONS := {
    Complement IS_COMPONENT_OF SetTheory,
    Complement IS_SUBTYPE_OF SetOperation,
    Complement DEPENDS_ON UniversalSet,
    Complement OPERATES_ON Set,
    Complement RETURNS Set,
    ComplementResult SUBSET_OF UniversalSet,
    ComplementResult DISJOINT_WITH TargetSet,
    Complement INTERACTS_WITH Union THROUGH DeMorganLaw,
    Complement INTERACTS_WITH Intersection THROUGH DeMorganLaw,
    RelativeComplement IS_SUBTYPE_OF SetDifference,
    Complement IS_SPECIAL_CASE_OF RelativeComplement
}

DEFINE SEMANTIC_EQUIVALENCE := {
    "complement" <=> Complement WHEN set_context=TRUE,
    "set complement" <=> Complement,
    "complement of A" <=> Complement(A),
    "A complement" <=> Complement(A),
    "Aᶜ" <=> Complement(A),
    "relative complement" <=> RelativeComplement,
    "set difference" <=> RelativeComplement WHEN expression="A\\B"
}

DEFINE AI_CONSUMPTION := {
    purpose: "machine-readable representation and evaluation of set complements",
    preserve_target_set_identity: TRUE,
    preserve_reference_universe: TRUE,
    require_universe_context_for_complement: TRUE,
    distinguish_complement_from_relative_difference: TRUE,
    apply_membership_negation_only_within_universe: TRUE,
    apply_involution: TRUE,
    apply_subset_reversal: TRUE,
    apply_DeMorgan_laws: TRUE,
    detect_universe_dependence: TRUE,
    reasoning_mode: "symbolic + extensional + context-aware + algebraic"
}

DEFINE CONFIDENCE := {
    ontology_identity: 1.00,
    formal_complement_detection: 0.99,
    universe_resolution_explicit: 0.99,
    complement_membership_evaluation: 0.99,
    involution_reasoning: 1.00,
    DeMorgan_reasoning: 0.99,
    relative_complement_mapping: 0.99,
    implicit_universe_resolution: context_dependent
}

DEFINE UPDATE_MODEL := {
    version: "1.0",
    update_policy: "append-compatible",
    immutable_core: {
        universe_relative_definition,
        membership_negation_within_universe,
        disjointness,
        exhaustion,
        involution,
        empty_universe_identities,
        DeMorgan_laws
    },
    extensible_layers: {
        Boolean_algebra_complement,
        relative_complements,
        logical_negation_correspondence,
        topology_complements,
        measure_theoretic_complements,
        computational_set_complement
    }
}

DEFINE FLOW_HASH_INPUT :=
    CanonicalURL
    + DOMAIN
    + ENTITY_TYPES
    + RELATION_TYPES
    + RULES
    + COMPLEMENT_MODEL
    + RELATIVE_COMPLEMENT_MODEL
    + COMPLEMENT_IDENTITIES
    + DE_MORGAN_MODEL
    + SEMANTIC_PIPELINE
    + REASONING_PIPELINE
    + COMPLEMENT_GRAPH_MODEL
    + ONTOLOGY_RELATIONS
    + AI_CONSUMPTION
    + UPDATE_MODEL

FLOW_HASH := HASH(FLOW_HASH_INPUT)

RETURN {
    ontology: ComplementOntology,
    canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/set-theory/complement-ontology.html",
    parent_domain: SetTheory,
    semantic_domain: MathematicalLogic,
    primary_entity: Complement,
    universe_sensitive: TRUE,
    extensional_reasoning_enabled: TRUE,
    algebraic_reasoning_enabled: TRUE,
    machine_consumable: TRUE,
    recursive_reasoning_enabled: TRUE,
    flow_complete: TRUE
}

END_FLOW