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# Bijective function

(Redirected from Bijection)

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## Contents

In mathematics, an invertible function, also known as a bijective function or simply a bijection is a function that establishes a one-to-one correspondence between elements of two given sets. Loosely speaking, all elements of the sets can be matched up in pairs so that each element of one set has its unique counterpart in the second set. A bijective function from a set X to itself is also called a permutation of the set X.

More formally, a function  from set  to set  is called a bijection if and only if for each  in  there exists exactly one  in  such that .

The most important property of a bijective function is the existence of an inverse function which undoes the operation of the function. These functions can then be viewed as dictionaries by which one can translate information from the domain to the codomain and back again. The existence of an inverse function often forces the domain and codomain to have common properties.

## Examples

• The function from set  to set  defined by the formula  is a bijection.
• A less obvious example is the function  from the set  of all pairs (x,y) of positive integers to the set of all positive integers given by formula .
• The function  is a bijection.

## Composition

If  and  are bijections than so is their composition .

A function  is a bijective function if and only if there exists function  such that their compositions  and  are identity functions on relevant sets. In this case we call function  an inverse function of  and denote it by .

## Bijections and the concept of cardinality

Two finite sets have the same number of elements if and only if there exists a bijection from one set to another. Georg Cantor generalized this simple observation to infinite sets and introduced the concept of cardinality of a set. We say that two set are equinumerous (sometimes also equipotent or equipollent) if there exists a bijection from one set to another. If this is the case, we say the sets have the same cardinality or the same cardinal number. Cardinality can be thought of as a generalization of number of elements of finite sets.

## Some more examples

1. A function is a bijection iff it is both an injection and a surjection.
2. The quadratic function  is neither injection nor surjection, hence is not bijection. However if we change its domain and codomain to the set  than the function becomes bijective and the inverse function  exists. This procedure is very common in mathematics, especially in calculus.
3. A continuous function from the closed interval  in the real line to closed interval  is bijection if and only if is monotonic function with f(a) = c and f(b) = d.