injectivity

Injectivity, also known as one-to-one, is a property of a function f: A → B defined by the condition that f(a1) = f(a2) implies a1 = a2 for all a1, a2 in A. Equivalently, distinct inputs map to distinct outputs, so the restriction of f to its domain A is a bijection from A onto its image f(A).

Consequences and related ideas: An injective function has a unique preimage for each element of its image. Elements of B that are not in the image have no preimage. If A and B are finite, injectivity implies |A| ≤ |B|; if |A| = |B|, injectivity also implies surjectivity, hence bijectivity. The function f: A → f(A) is always a bijection.

Examples: 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 constant function f(x) = c is injective only when the domain has a single element.

Inverse and composition: If there exists a function g: B → A with g∘f = id_A, then f is injective. In general, a function injective on its domain has a left inverse on its image. If f: A → B and g: B → C are injective, then their composition g∘f is injective; and if g∘f is injective, then f is injective.

In linear algebra and algebra, injectivity is often characterized by a trivial kernel: a linear map T is injective iff ker(T) = {0}. Injectivity is distinct from surjectivity; a bijection is both injective and surjective and therefore invertible.