injektiv

Injektiv, often translated as injective or one-to-one, is a fundamental concept in mathematics used to describe a type of function.

A function f: A → B is injective if different inputs map to different outputs. Formally, for all x, y in A, f(x) = f(y) implies x = y. Equivalently, if x ≠ y then f(x) ≠ f(y). This means that no two distinct elements of the domain share the same image.

Examples illustrate the idea. The function f(x) = 2x from the integers to the integers is injective. The function f(x) = x^2 from the real numbers to the real numbers is not injective, since f(1) = f(-1). The exponential function f(x) = e^x from the real numbers to (0, ∞) is injective.

Key properties include that the restriction of f to its image is a bijection between A and Im(f). Therefore f has a left inverse defined on Im(f). If the domain and codomain are finite and f is injective, then the size of the domain does not exceed the size of the codomain: |A| ≤ |B|.

Composition preserves injectivity under certain conditions. If f: A → B and g: B → C are injective, then the composition g ∘ f: A → C is injective. More generally, injective maps are preserved by composition when each map is injective on the relevant domain.

Related concepts include surjectivity (onto) and bijectivity (both injective and surjective). In category theory, injective maps correspond to monomorphisms in the category of sets.