> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/260-function-theory/abjections/bijection.md). # Bijective function A bijective function is a function that is both injective and surjective. This means that a bijective function imposes equinumerosity between its domain, codomain and range. > bijection: domain === range === codomain That's why another name for a bijective function is a *one-to-one correspondence* (it should not be confused with the term "one-to-one function" that refers to an injective function). Bijective functions are also referred to as *bijections* (this term is not exclusive to functions). *** correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A mapping that is both injective and surjective is bijective. More precisely, a mapping that is first injection, then surjection, is a bijection. That's because surjection has a good and a bad property: its good property ensures that the entire codomain is involved in the mapping, but its bad property means that some elements of the codomain are double-mapped. This also means that an injection can be upgraded to a bijection, but surjection cannot. * Bijection means a mapping is both one-to-one and onto. It is a perfect mapping from one element of the domain to one element in the codomain, with the entire codomain involved in the mapping. * Bijective mapping: `dom(f) = cod(f) = ran(f)` * Bijective mapping is fully invertable; an `f` has an inverse mapping, `f⁻¹`, `f(a)=b -> ∀b ∈ B. f⁻¹(b)=a`. * If the codomain of an injection `f` is shrinked to be equal to the range of `f`, the `f` would be bijective. So, an injection can be "upgraded" to a bijection, which cannot be done for surjection.