Bijective function

https://en.wikipedia.org/wiki/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).

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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.