Bijection

A bijection is a function that is both injective and surjective. It pairs each element of its codomain with exactly one element of its domain, so an inverse function exists.

Function: A function assigns each element of a domain exactly one element of a codomain. Bijections are a special kind of function.
Inverse function: An inverse function reverses a function's input-output assignments. A bijection has an inverse defined on its entire codomain.
Equipotence: Two sets are equipotent when a bijection exists between them. A bijection gives the standard criterion for sets to have equal size.
Many-to-one function: A many-to-one function maps at least two distinct inputs to the same output. It violates injectivity, so it cannot be bijective.
Injective function: A function is injective when distinct domain elements have distinct images. This property prevents two domain elements from sharing an image.
Composition of functions: Composition combines functions by applying one function's output as another's input. Composing bijections produces another bijection.
Cardinality: The cardinality of a set measures its number of elements, including infinite sizes. Bijections let cardinalities be compared without counting elements directly.
Partial function: A partial function is defined only on part of a specified set of possible inputs. A bijection is defined on every element of its stated domain.
Surjective function: A function is surjective when every element of its codomain is the image of some domain element. This property ensures the function reaches the entire codomain.
Identity function: An identity function maps every element of a set to itself. Composing a bijection with its inverse gives identity functions.

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