Surjective function (onto): definition, examples, properties, and distinctions

Formal definition and basic properties

Formally, f: A → B is surjective if for every b in B there exists an a in A with f(a) = b. Equivalently, the set f(A) equals B. Surjectivity is a property of the function together with its specified codomain: the same rule can be surjective or not depending on what B is taken to be.

• Image equals codomain: f(A) = B.
• Right-inverse characterization: f is surjective precisely when there exists a function g: B → A such that f∘g is the identity on B.
• Composition: if g∘f is surjective then g must be surjective; f need not be.

Examples and non-examples

Common examples of surjective maps include linear surjections between vector spaces where the image spans the target, and the exponential map from real numbers to positive real numbers. Simple non-examples are constant maps to a codomain with more than one element, or the inclusion of a proper subset as the codomain. A few illustrative cases:

• Surjective: f: R → (0,∞), f(x)=e^x is onto the positive real numbers.
• Surjective: any bijection is surjective by definition.
• Not surjective: f: R → R, f(x)=e^x is not onto R because negative numbers are excluded from the codomain R.
• Finite sets: a function from a set with at least as many elements as the codomain can be surjective; for equal finite sizes, surjectivity is equivalent to injectivity.

Equivalent characterizations and tests

Aside from the direct "for every b there exists a" definition, useful equivalent viewpoints include the existence of a right inverse (a section) and categorical notions: in category theory, surjections correspond to epimorphisms in familiar concrete categories like Set. For finite domains and codomains of equal cardinality, injective, surjective, and bijective are all equivalent. For infinite sets, care is needed: a surjection from one infinite set onto another can coexist with injections in both directions without implying equality of size in the naive sense.

Uses, importance, and connections

Surjective functions appear throughout mathematics: solving equations amounts to finding preimages of values, quotient constructions use surjections to identify elements, and many classification results are expressed as surjective maps onto parameter spaces. In algebra, homomorphisms that are onto define quotient structures; in topology, continuous surjections describe space-filling maps and quotient topologies. For further comparisons with other map types, see introductions to injective and bijective maps.

History and terminology

The term surjection (from the French prefix sur-, meaning "over" or "onto") and its companions injection and bijection were standardized in 20th-century mathematics as part of efforts to formalize set theory and mappings; the collective pseudonym who popularized this notation is associated with advanced mathematical texts. The standard vocabulary and many examples are discussed in basic texts on functions and set theory; a compact discussion of codomain and range distinctions appears in resources focusing on the definition of function codomain and range.

Notable distinctions

It is important to distinguish between the image (the actual set of outputs) and the codomain (the set designated as targets). Changing the codomain can change whether a map is surjective. Also note that surjectivity alone does not guarantee invertibility; a two-sided inverse exists only for bijections. For practical verification, construct explicit preimages for every element of the codomain or exhibit a right inverse when possible.