Surjective function

A function is surjective when every element of its codomain is the output of at least one input. Its image therefore equals its codomain.

Function (mathematics): A relation that assigns each element of a domain exactly one element of a codomain. Surjectivity is a property of functions, defined by which codomain elements their outputs reach.
Preimage: The set of inputs that a function maps to a specified output or subset of its codomain. Surjectivity requires each codomain element to have a nonempty preimage.
Injective function: A function that maps distinct inputs to distinct outputs. Injectivity concerns repeated outputs, while surjectivity concerns uncovered codomain elements.
Surjective linear map: A linear transformation whose image equals its codomain vector space. In linear algebra, surjectivity means every target vector has a preimage.
Composition of functions: The operation of applying one function to the output of another. A composition of surjective functions is surjective, since each stage reaches its full codomain.
Domain of a function: The set of inputs on which a function is defined. Surjectivity asks whether inputs from this set reach every codomain element.
Existential quantification: A logical quantifier asserting that at least one object satisfies a stated condition. The definition says that for each codomain element, some domain input maps to it.
Bijective function: A function that is both injective and surjective. Adding injectivity to surjectivity gives a one-to-one correspondence between domain and codomain.
Rank–nullity theorem: A theorem equating a linear map's domain dimension with the sum of its rank and nullity. It turns surjectivity of finite-dimensional linear maps into a dimension condition.
Cantor's theorem: The theorem that no set has a surjection onto its power set. It uses the limits of surjective maps to establish that power sets are strictly larger.

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