Noninjective

Noninjective refers to a function that is not injective. A function f from a set A to a set B is injective (one-to-one) if distinct inputs map to distinct outputs: f(a1) = f(a2) implies a1 = a2. If this implication fails for some pair of distinct elements, the function is noninjective.

Examples illustrate the concept. The function f(n) = n^2 from the integers to the integers is noninjective because f(3) = f(-3) although 3 ≠ -3. The trigonometric function f(x) = cos(x) from the real numbers to the real numbers is also noninjective, since multiple x values yield the same cosine value (for instance, cos(0) = cos(2π)).

Noninjectivity has several consequences. It means there is no unique inverse on the whole codomain; a left inverse, which would satisfy g(f(x)) = x for all x, cannot exist unless f is injective. A function can be surjective (onto) without being injective, and in that case it may possess a right inverse defined on the codomain (or on its image). Conversely, a function can be injective but not surjective.

In broader contexts, noninjective maps are common and often involve collisions or identifications. For example, hash functions are typically noninjective since many inputs can produce the same hash value. In algebra and category theory, injectivity relates to the existence of certain inverses or universal properties, while noninjectivity indicates the presence of identifications among distinct elements.