Bijective function: Difference between revisions

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imported>Wojciech Świderski
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For example, a function from set <math>\{1,2,3,4\}</math> to set <math>\{10,11,12,13\}</math> defined by formula <math>f(x)=x+9</math> is bijection.
For example, a function from set <math>\{1,2,3,4\}</math> to set <math>\{10,11,12,13\}</math> defined by formula <math>f(x)=x+9</math> is bijection.


Less obvious example is function <math>f</math> from the set <math>X=\{(x,y)\}</math> of all ''pairs'' (x,y) of [[integer|positive integers]] to the set of all positive integers given by formula <math>f(x,y)=2^x\cdot (2y-1)</math>.
Less obvious example is function <math>f</math> from the set <math>X=\{(x,y)\}</math> of all ''pairs'' (x,y) of [[integer|positive integers]] to the set of all positive integers given by formula <math>f(x,y)=2^{x-1}\cdot (2y-1)</math>.


Function <math>\tan\colon(-\frac{\pi}{2},\frac{\pi}{2})\to R</math> is another example of bijection.
Function <math>\tan\colon(-\frac{\pi}{2},\frac{\pi}{2})\to R</math> is another example of bijection.

Revision as of 11:15, 13 July 2008

Bijective function is a function that establishes a one-to-one correspondence between elements of two given sets. Loosely speaking, all elements of those sets can be matched up in pairs so that each element of one set has its counterpart in the second set.

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 .

For example, a function from set to set defined by formula is bijection.

Less obvious example is function from the set of all pairs (x,y) of positive integers to the set of all positive integers given by formula .

Function is another example of bijection.

A bijective function from a set X to itself is also called a permutation of the set X.

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 set have the same cardinality or the same cardinal number. Cardinal number can be thought of as a generalization of number of elements of final set.

Some more examples

  1. A function is a bijection is both injection and surjection.
  2. The quadratic function <maht>R\to R: x\mapsto x^2</math> 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 closed interval of real line onto closed interval is bijection if and only if is monotonic funtion.