Nonbijective

Nonbijective describes a function that is not bijective. In mathematics, a function f from a set A to a set B is bijective if it is both injective (one-to-one) and surjective (onto). A nonbijective function fails to meet one or both of these criteria.

Nonbijective functions can fail to be injective, fail to be surjective, or fail at both. If f is not injective, two or more elements of A map to the same element of B, making the function many-to-one. For example, f: {1,2,3} → {a,b} with f(1)=a, f(2)=a, f(3)=b is surjective but not injective. If f is not surjective, some elements of B have no preimage; for instance, g: {1,2} → {a,b,c} with g(1)=a, g(2)=b is injective but not surjective. A function can also be both non-injective and non-surjective, such as h: {1,2} → {a} with h(1)=a, h(2)=a.

The concept is central to invertibility. Bijective functions have a two-sided inverse, meaning there exists a function f^{-1}: B → A with f^{-1}(f(x)) = x for all x in A and f(f^{-1}(y)) = y for all y in B. Nonbijective functions lack such an inverse and cannot be reversed uniquely. In finite sets of equal size, bijectivity is equivalent to both injectivity and surjectivity; otherwise, the function is nonbijective. Related notions include injective (one-to-one), surjective (onto), and isomorphisms in category theory, where bijections correspond to isomorphisms in the category of sets.