Bijective function

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 $f$ from set $X$ to set $Y$ is called a bijection if and only if for each $y$ in $Y$ there exists exactly one $x$ in $X$ such that $f(x)=y$.

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 $\{1,2,3,4\}$ to set $\{10,11,12,13\}$ defined by the formula $f(x)=x+9$ is a bijection.
• A less obvious example is the function $f$ from the set $X=\{(x,y)\}$ of all pairs (x,y) of positive integers to the set of all positive integers given by formula $f(x,y)=2^{x-1}\cdot (2y-1)$.
• The function $\tan \colon (-{\frac {\pi }{2}},{\frac {\pi }{2}})\to R$ is a bijection.

Composition

If $f\colon X\to Y$ and $g\colon Y\to Z$ are bijections than so is their composition $g\circ f\colon X\to Z$.

A function $f\colon X\to Y$ is a bijective function if and only if there exists function $g\colon Y\to X$ such that their compositions $g\circ f$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\circ g} are identity functions on relevant sets. In this case we call function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} an inverse function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and denote it by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} .

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R\to R: x\mapsto x^2} is neither injection nor surjection, hence is not bijection. However if we change its domain and codomain to the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,+\infty)} than the function becomes bijective and the inverse function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt\colon [0,+\infty)\to[0,+\infty),\ x\mapsto \sqrt{x}} exists. This procedure is very common in mathematics, especially in calculus.
3. A continuous function from the closed interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} in the real line to closed interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [c,d]} is bijection if and only if is monotonic function with f(a) = c and f(b) = d.