nonsurjectivity

Nonsurjectivity refers to a property of a function that is not surjective. A function f: A → B is surjective (onto) when every element of the codomain B has at least one preimage in the domain A, meaning the image of f equals the entire codomain B. A function is nonsurjective if this condition fails, i.e., the image of f is a proper subset of B.

The essential idea is that the range, or image, of a function is the set of all outputs produced by inputs from the domain. If the image does not cover all of the codomain, the function is not onto. This can occur even if the function is well-defined for every input in A.

Examples help illustrate nonsurjectivity. The function f: {1, 2} → {a, b, c} defined by f(1) = a and f(2) = b is not surjective because c has no preimage. The function g: R → R given by g(x) = x^2 is not surjective, since negative numbers are not squares and thus have no preimage.

Nonsurjectivity is a distinct property from noninjectivity. A function can be non-surjective and injective (one-to-one), non-injective and surjective, or neither. In finite settings, surjectivity imposes a relationship between the sizes of the domain and codomain: if f is surjective, the image has size equal to the codomain; consequently, if |A| < |B|, no function A → B can be surjective. Conversely, bijectivity requires both surjectivity and injectivity. In general, a surjective function admits a right inverse (assuming an appropriate form of the axiom of choice).