For example, consider the function succ, defined from the set of integers to , that to each integer x associates the integer succ(x) = x + 1. For another example, consider the function sumdif that to each pair (x,y) of real numbers associates the pair sumdif(x,y) = (x+y, x-y).
A bijective function is also called a bijection or permutation. The latter is more commonly used when X = Y. It should be noted that one-to-one function means one-to-one correspondence (i.e., bijection) to some authors, but injection to others. The set of all bijections from X to Y is denoted as XY.
Bijective functions play a fundamental role in many areas of mathematics, for instance in the definition of isomorphism (and related concepts such as homeomorphism and diffeomorphism), permutation group, projective map, and many others.
Composition and inverses
A function f is bijective if and only if its inverse relation f-1 is a function. In that case, f-1 is a bijection.
The composition gf of two bijections f XY and g YZ is a bijection. The inverse of gf is (gf)-1 = (f-1)(g-1).
On the other hand, if the composition g o f of two functions is bijective, we can only say that f is injective and g is surjective.
A relation f from X to Y is a bijective function if and only if there exists another relation g from Y to X such that gf is the identity function on X, and fg is the identity function on Y. It is important to say that having the same cardinality for two sets is a must.
Bijections and cardinality
If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Indeed, in axiomatic set theory, this is taken as the very definition of "same number of elements", and generalising this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets.
Examples and counterexamples
- For any set X, the identity function idX from X to X, defined by idX(x) = x, is bijective.
- The function f from the real line R to R defined by f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y - 1)/2 such that f(x) = y.
- The exponential function g : R R, with g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) = -1, showing that g is not surjective. However if the codomain is changed to be the positive real numbers R+ = (0,+∞), then g becomes bijective; its inverse is the natural logarithm function ln.
- The function h : R [0,+∞) with h(x) = x² is not bijective: for instance, h(-1) = h(+1) = 1, showing that h is not injective. However, if the domain too is changed to [0,+∞), then h becomes bijective; its inverse is the positive square root function.
- A function f from the real line R to R is bijective if and only if its plot is intersected by any horizontal line at exactly one point.
- is not a bijection because -1, 0, and +1 are all in the domain and all map to 0.
- is not a bijection because π / 3 and are both in the domain and both map to .
- If X is a set, then the bijective functions from X to itself, together with the operation of functional composition (o), form a group, the symmetric group of X, which is denoted variously by S(X), SX, or X! (the last read "X factorial").
- For subset A of the domain and subset B of the codomain we
|f(A)| == |A|, and |f-1(B)| == |B|.
(Note: a one-to-one function is injective, but may fail to be surjective, while a one-to-one correspondence is both injective and surjective.)