injective

Injective is a term used in mathematics to describe a function, map, or morphism that preserves distinctness: if f: X → Y and f(x1) = f(x2) then x1 = x2. Equivalently, different inputs yield different outputs, so f restricted to its domain is a bijection onto its image. The image f[X] is a subset of Y, and the map f: X → f[X] is a bijection.

Examples include f(n) = 2n from the integers to the integers, which is injective but not surjective; the inclusion map i: A ⊆ B is injective; and the exponential function exp: R → (0, ∞) is injective. These illustrate that a function can be injective without being surjective.

Injectivity has several standard consequences. An injective function has a left inverse on its image: there exists g: Y → X with g ∘ f = id_X, obtained by mapping each y in the image f[X] to the unique x with f(x) = y and assigning arbitrary values elsewhere. The composition of injections is injective, and restricting the domain of an injective function to a subset preserves injectivity. If f is injective, X can be identified with its image f[X] via the natural bijection.

In linear algebra, a linear map T: V → W is injective iff ker(T) = {0}; equivalently, rank(T) = dim(V) when these dimensions are finite. In abstract algebra and category theory, “injective” often refers to a monomorphism, a morphism with a left-cancellation property. Injectivity is contrasted with surjectivity and bijectivity, and underpins many embedding constructions across mathematics.