Injective Function Ontology

FLOW ALGEBRA
INJECTIVE FUNCTION ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Function and Mapping Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/function-and-mapping-entities/injective-function-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13
Status: Formal mathematical synthesis; source webpage content not independently verified.

01. ONTOLOGY IDENTITY
ontology_id: kireth.math.function.injective
entity_type: InjectiveFunction
parent: Function
aliases: [one-to-one function, injection, injective mapping]
meaning: A function whose distinct inputs always have distinct outputs.

02. TYPE SIGNATURE
sets: A, B
function: f : A -> B
predicate: Injective(f)
preconditions: f is single-valued and total on A

03. DEFINING AXIOM
Injective(f) <=> forall x,y in A: f(x)=f(y) => x=y.
Equivalent: forall x,y in A: x!=y => f(x)!=f(y).
A function's injectivity is relative to its specified domain.

04. DOMAIN AND CODOMAIN
Domain(f)=A
Codomain(f)=B
Image(f)={f(x) | x in A} subseteq B
Injectivity does not imply Image(f)=B.

05. FIBRE CHARACTERISATION
Fibre(f,b)={a in A | f(a)=b}.
Injective(f) <=> forall b in B: |Fibre(f,b)| <= 1.
Empty fibres are permitted.

06. GRAPH CHARACTERISATION
Graph(f)={(a,f(a)) | a in A} subseteq A x B.
Injective(f) <=> each second coordinate appears in at most one graph pair.

07. LEFT INVERSE
If A is nonempty and f:A->B is injective, a left inverse g:B->A exists under classical set theory with a chosen element of A.
g(f(a))=a for all a in A.
Without an assumption that A is nonempty, the empty-domain/nonempty-codomain case has no left inverse B->A.
A canonical inverse f^{-1}:Image(f)->A always exists for injective f.

08. RESTRICTED INVERSE
inverse_on_image: f^{-1}(f(a))=a.
for b in Image(f): f(f^{-1}(b))=b.
This is a bijection between A and Image(f).

09. COMPOSITION
If f:A->B and g:B->C are injective, then g o f is injective.
If g o f is injective, then f is injective.
If f is surjective and g o f injective, then g is injective.

10. RESTRICTION
If f:A->B is injective and S subseteq A, then f|S:S->B is injective.
A noninjective function may become injective after restricting its domain.

11. FINITE SET CARDINALITY
For finite A,B, injective f:A->B implies |A|<=|B|.
For finite A and B with |A|=|B|, injective <=> surjective <=> bijective.
Pigeonhole principle: |A|>|B| implies no injection A->B.

12. INFINITE SET CARDINALITY
A injects into B is a comparison of cardinalities: |A|<=|B|.
Cantor-Schroeder-Bernstein: injections A->B and B->A imply a bijection A<->B.
Infinite injective maps need not be surjective, e.g. n->n+1 on N.

13. EMPTY AND SINGLETON CASES
Every function from the empty set is injective (vacuously).
Every function from a singleton domain is injective.
A function from an empty domain to an empty codomain is bijective.

14. EXAMPLES
f:Z->Z, f(n)=2n: injective, not surjective.
f:R->R, f(x)=x^3: bijective, hence injective.
f:R->R, f(x)=x^2: not injective because f(1)=f(-1).
f:[0,infinity)->R, f(x)=x^2: injective, not surjective.

15. LINEAR ALGEBRA
For linear T:V->W, Injective(T) <=> ker(T)={0}.
For matrix M defining T(x)=Mx over a field, injective <=> nullity(M)=0.
For finite-dimensional V, rank(T)=dim(V) when T is injective.

16. ORDER AND CONTINUITY
Strictly monotone real functions are injective.
A continuous injective function on an interval is strictly monotone.
A function need not be continuous or monotone to be injective.

17. SEMANTIC RELATIONSHIPS
InjectiveFunction is_a Function.
InjectiveFunction has_property UniquePreimageForImageElements.
BijectiveFunction implies InjectiveFunction.
InjectiveFunction is_not_equivalent_to SurjectiveFunction.
InjectiveFunction enables InverseOnImage.

18. KNOWLEDGE GRAPH TRIPLES
(InjectiveFunction, subClassOf, Function)
(InjectiveFunction, definedBy, DistinctInputsDistinctOutputs)
(InjectiveFunction, characterisedBy, FibreCardinalityAtMostOne)
(InjectiveFunction, permits, LeftInverseUnderConditions)
(InjectiveFunction, composesWith, InjectiveFunction)
(InjectiveFunction, relatedTo, CardinalityComparison)

19. INFERENCE RULES
R1: Function(f) and forall x,y in Domain(f), f(x)=f(y)=>x=y -> Injective(f).
R2: Injective(f) and Injective(g) and Codomain(f)=Domain(g) -> Injective(g o f).
R3: Injective(f) and S subseteq Domain(f) -> Injective(restrict(f,S)).
R4: Injective(f) and Surjective(f) -> Bijective(f).
R5: exists x!=y with f(x)=f(y) -> NotInjective(f).
R6: Injective(f) -> InvertibleOnImage(f).

20. VALIDATION RULES
V1: Check that f is a function with declared domain and codomain.
V2: Check domain membership before evaluating equality tests.
V3: Distinguish codomain from image.
V4: One counterexample pair disproves injectivity.
V5: Finite exhaustive checks establish injectivity only when domain is fully enumerated.
V6: Numerical sampling cannot prove injectivity over an infinite domain.
V7: Treat floating-point near-equality as approximate evidence, not exact equality.

21. COUNTEREXAMPLE CERTIFICATE
certificate_type: NonInjectiveWitness
fields: [function_id, x, y, f_x, f_y, domain_evidence]
valid_when: x!=y and f_x=f_y with x,y in Domain(f).

22. PROOF CERTIFICATE
certificate_type: InjectivityProof
methods: [direct_equality_argument, contrapositive, inverse_on_image, kernel_test, strict_monotonicity, exhaustive_finite_domain]
requirements: proof_method, assumptions, domain, result, provenance.

23. SEMANTIC EQUIVALENCE
"one-to-one function" ~= "injective function" in mathematical context.
"inverse function" is not equivalent to "injective function" without specifying image/codomain.
"distinct outputs" alone is ambiguous unless quantified over distinct inputs.

24. CONFIDENCE
confidence.mathematical_axiom: 1.0
confidence.proven_instance: 1.0 when a valid formal proof is supplied
confidence.finite_exhaustive: 1.0 under complete exact enumeration
confidence.numerical_sample: heuristic_only
confidence.unverified_claim: unknown

25. FLOW HASH
hash_algorithm: SHA-256
canonicalisation: UTF-8, LF newlines, stable key order, normalised notation
hash_scope: ontology body excluding hash value and mutable update timestamps
FLOW_HASH: compute_at_publication; do_not_fabricate

26. UPDATE MODEL
versioning: semantic_versioning
change_triggers: [definition_revision, relation_revision, proof_rule_revision, validation_revision]
update_steps: [validate, normalise, compare, version, recompute_hash, publish]
retain_provenance: true

27. AI CONSUMPTION
entity_resolution: match aliases to ontology_id.
reasoning_mode: symbolic_first.
return_fields: [entity_type, domain, codomain, claim, proof_or_counterexample, confidence, provenance]
error_policy: do_not_infer_injectivity_from_samples_alone.
uncertainty_policy: label unproved claims explicitly.

28. TEST CASES
T1: f(n)=2n on Z -> injective.
T2: f(x)=x^2 on R -> not injective; witness (-1,1).
T3: f(x)=x^2 on nonnegative reals -> injective.
T4: f(n)=n+1 on N -> injective, not surjective (for N={0,1,...}).
T5: empty function empty->B -> injective for any B.
T6: injective f and g -> g o f injective where types match.

29. TERMINATION
status: COMPLETE
integrity: all numbered sections present
end_of_flow: true