Bijective function

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 $f$ 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 $f(x)=y$.

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

Less obvious example is 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 $f(x,y)=2^{x-1}\cdot (2y-1)$.

Function $\tan \colon (-{\frac {\pi }{2}},{\frac {\pi }{2}})\to R$ 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 $f\colon X\to Y$ 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 $f\circ g$ are identity functions on relevant sets. In this case we call function $g$ an inverse function of $f$ and denote it by $f^{-1}$.

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 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 iff it is both an injection and a surjection.
2. The quadratic function $R\to R:x\mapsto x^{2}$ is neither injection nor surjection, hence is not bijection. However if we change its domain and codomain to the set $[0,+\infty )$ than the function becomes bijective and the inverse function ${\sqrt {\colon }}[0,+\infty )\to [0,+\infty ),\ x\mapsto {\sqrt {x}}$ exists. This procedure is very common in mathematics, especially in calculus.
3. A continuous function from the closed interval $[a,b]$ in the real line to closed interval $[c,d]$ is bijection if and only if is monotonic funtion with f(a) = c and f(b) = d.