Sample Space Ontology

FLOW ALGEBRA
TITLE: Sample Space Ontology
DOMAIN: Discrete Mathematics / Discrete Probability
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/discrete-probability/sample-space-ontology-2.html
VERSION: 1.0
LANGUAGE: en-GB
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
SOURCE_STATUS: Mathematical synthesis; live page not independently verified.

1. PURPOSE
Represent a discrete sample space as the set of all possible outcomes of a random experiment, with explicit typing, event relationships, probability semantics, and machine-checkable constraints.

2. TYPED ENTITIES
RandomExperiment := {id: String, description: String}
Outcome := {id: String, experiment_id: String}
SampleSpace := {experiment_id: String, omega: Set[Outcome], cardinality: Nat | CountablyInfinite}
Event := {sample_space_id: String, members: Subset[Outcome]}
ProbabilityMassFunction := {sample_space_id: String, p: Outcome -> Real[0,1]}
ProbabilityMeasure := {sample_space_id: String, P: Event -> Real[0,1]}
Partition := {sample_space_id: String, cells: Set[Event]}
RandomVariable := {sample_space_id: String, X: Outcome -> Value}

3. CORE DEFINITIONS
sample_space(E) = Omega(E)
possible_outcome(w,E) iff w in Omega(E)
event(A,E) iff A subseteq Omega(E)
certain_event(E) = Omega(E)
impossible_event(E) = empty_set
complement(A,E) = Omega(E) \ A
finite_discrete(E) iff |Omega(E)| is a finite natural number
countably_infinite_discrete(E) iff Omega(E) is countably infinite

4. FLOW OPERATIONS
flow enumerate_outcomes(E) -> Omega(E)
  pre: E is a defined experiment with enumerable outcomes
  post: each distinct possible elementary outcome occurs exactly once

flow construct_event(E,predicate) -> A
  A := {w in Omega(E) : predicate(w)}
  assert A subseteq Omega(E)

flow union_events(A,B) -> A union B
  pre: same_sample_space(A,B)

flow intersect_events(A,B) -> A intersection B
  pre: same_sample_space(A,B)

flow complement_event(A,E) -> Omega(E) \ A
  pre: A subseteq Omega(E)

flow assign_probability(E,p) -> P
  pre: for all w in Omega(E), 0 <= p(w) <= 1
  pre: sum_{w in Omega(E)} p(w) = 1
  P(A) := sum_{w in A} p(w)

5. AXIOMS AND LAWS
A subseteq Omega
P(empty_set) = 0
P(Omega) = 1
P(A) >= 0
P(A union B) = P(A) + P(B) - P(A intersection B)
P(Omega \ A) = 1 - P(A)
If A intersection B = empty_set then P(A union B) = P(A) + P(B)
For countably disjoint events (A_i), P(union_i A_i) = sum_i P(A_i)
If A subseteq B then P(A) <= P(B)
For uniform finite Omega: P(A) = |A| / |Omega|, provided |Omega| > 0

6. RELATIONSHIPS
RandomExperiment --has_sample_space--> SampleSpace
SampleSpace --contains--> Outcome
Event --is_subset_of--> SampleSpace
Outcome --belongs_to--> Event [conditional]
Event --has_complement--> Event
ProbabilityMassFunction --assigns_mass_to--> Outcome
ProbabilityMeasure --assigns_probability_to--> Event
Partition --partitions--> SampleSpace
RandomVariable --maps_outcome_to--> Value

7. INFERENCE RULES
R1: w in A and A subseteq Omega => w in Omega
R2: A subseteq Omega => Omega \ A is an Event
R3: A subseteq B => P(A) <= P(B)
R4: A intersection B = empty_set => P(A union B) = P(A) + P(B)
R5: partition(C,Omega) => union(C)=Omega and distinct cells are disjoint
R6: finite(Omega) and equal_mass(p) and |Omega|>0 => p(w)=1/|Omega|

8. EXAMPLES
Fair die:
  Omega = {1,2,3,4,5,6}
  A = {2,4,6}
  P(A) = 3/6 = 1/2
Two ordered coin tosses:
  Omega = {HH,HT,TH,TT}
  B = {HT,TH}
  P(B) = 1/2 if the four outcomes are equiprobable
Countable example:
  Omega = {1,2,3,...}
  p(n) = 2^(-n) for n>=1
  sum_{n>=1} p(n) = 1

9. VALIDATION
V1: Outcome identifiers are unique within Omega.
V2: Every event contains only outcomes from its sample space.
V3: All probability masses are non-negative and normalised.
V4: No uniform distribution is assumed without evidence.
V5: Event operations require a common sample space or explicit embedding.
V6: A countably infinite sample space requires a convergent mass sum.
V7: An empty sample space cannot carry a normalised probability measure.
V8: Partition cells are non-empty, pairwise disjoint and cover Omega.

10. SEMANTIC_EQUIVALENCE
sample_space == outcome_space [within probability context]
elementary_outcome == sample_point
certain_event == whole_sample_space
impossible_event == empty_event
NOTE: event != outcome; an event is a set of outcomes.
NOTE: discrete != necessarily finite; countably infinite spaces are permitted.

11. CONFIDENCE
mathematical_definitions: 1.00
formal_axioms: 1.00
example_validity: 1.00
live_page_alignment: unverified
confidence_policy: Do not equate mathematical certainty with verification of the source page.

12. FLOW_HASH
algorithm: SHA-256
canonicalisation: UTF-8; LF line endings; exclude the FLOW_HASH value field
value: compute_on_ingestion

13. UPDATE_MODEL
version: 1.0
update_trigger: corrected definitions, extended examples, source revisions or schema changes
update_policy: preserve canonical identifiers; increment version; regenerate FLOW_HASH; record provenance

14. AI_CONSUMPTION
parse_mode: typed_flow_algebra
primary_entity: SampleSpace
reasoning_tasks: outcome enumeration, event construction, probability validation, discrete modelling
required_checks: V1,V2,V3,V4,V5,V6,V7,V8
output_expectation: typed entities, valid relations, proof trace, uncertainty disclosure
provenance: SOURCE_URL

END FLOW ALGEBRA