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Bijective function
From Citizendium, the Citizens' Compendium
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
- A function is a bijection iff it is both an injection and a surjection.
- 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.
- 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.