injektiver

Injektive, often written injektiv in German mathematics, is the adjective used to describe a type of mapping that is called injective in English. The noun form of the property is Injektivität. The term is used across algebra, analysis, and related fields to denote that a function preserves distinctness of inputs.

Definition and basic idea

A function f: A → B is injektiv if different elements of A map to different elements of B. Formally, f(x1) = f(x2) implies x1 = x2 for all x1, x2 in A. Equivalently, the preimage of every element of B contains at most one element of A. In finite contexts, this means the image f(A) has the same cardinality as A.

Key properties

Injectivity is preserved under composition: if f: A → B and g: B → C are injective, then g ∘ f: A → C is injective. If f is injective, it has a left inverse on its image, and f is invertible when restricted to A onto f(A). The concept contrasts with surjectivity (being onto) and bijectivity (being both injective and surjective).

Examples

The function f(n) = 2n from integers to integers is injective. The function f(n) = n mod 3 from integers to {0,1,2} is not injective, since different inputs can yield the same remainder. A simple injective mapping from a set {1,2,3} to {a,b,c} is 1→a, 2→b, 3→c.

Context and related concepts

In Set theory, injective functions are equivalent to monomorphisms. In German texts, the term Injektivität and its adjective form injektiv or injektiver are standard. Related concepts include bijection (one-to-one and onto) and surjection (onto). See also Monomorphismus, Bijektion, and Surjektion.