injectifs

Injectifs, or injective functions, are mathematical mappings that preserve distinctness: if x ≠ y are elements of the domain, then f(x) ≠ f(y) in the codomain. They are called one-to-one functions in many contexts.

Formally, a function f: A → B is injective if f(x) = f(y) implies x = y for all x, y in A. Equivalently, f is injective when each element of the codomain is the image of at most one element of the domain; equivalently, f has a left inverse on its image, and, in finite cases, |f(A)| = |A|.

Examples illustrate the idea. The function f(x) = 2x from the real numbers to the real numbers is injective, since distinct inputs produce distinct outputs. The function f(x) = x^2 from R to R is not injective because f(1) = f(-1). However, restricting the domain to [0, ∞) makes f(x) = x^2 injective on that domain.

Injectifs have consequences and uses across mathematics. They enable arguments about uniqueness and embeddings, such as proving that a structure can be faithfully represented inside another without collapsing distinct elements. In finite settings, injectivity is often used to infer that a mapping preserves cardinality. In computer science, injective mappings underpin hashing schemes and data encoding where distinct inputs must yield distinct outputs. In abstract algebra and category theory, injective notions relate to monomorphisms and left-cancellability.

See also: injective function, injection, surjective function, bijection, monomorphism. Etymology traces to Latin roots denoting “not throwing away” distinct inputs into identical outputs.