A bijective function.

In mathematics, a function f from a set X to a set Y is said to be bijective if and only if for every y in Y there is exactly one x in X such that f(x) = y. A bijection. ... A bijection. ... Euclid, detail from The School of Athens by Raphael. ... Partial plot of a function f. ... This article is about sets in mathematics. ... ↔ ⇔ ≡ logical symbols representing iff. ...

Said another way, f is bijective if and only if it is a one-to-one correspondence between those sets; i.e., both one-to-one (injective) and onto (surjective). In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...

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). The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

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. In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...

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. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ... A projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections. ...

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 also a bijection. In logic and mathematics, the inverse relation of a binary relation is the binary relation defined by . ...

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). In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

A bijective composition.

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. Image File history File links An illustration of two functions/mappings: the left is injective and non-surjective and the right is non-injective and surjective. ... Image File history File links An illustration of two functions/mappings: the left is injective and non-surjective and the right is non-injective and 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. Consequently, the sets have the same cardinality. An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

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. In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... ↔ ⇔ ≡ logical symbols representing iff. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... Infinity refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. ...

Examples and counterexamples

• For any set X, the identity function id_X from X to X, defined by id_X(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) = e^x, 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.
• 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 2π/3 are both in the domain and both map to (√3)/2.

An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In mathematics, the real line is simply the set of real numbers. ... The exponential function is one of the most important functions in mathematics. ... The natural logarithm is the logarithm to the base e, where e is equal to 2. ...

Properties

• 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.
• 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), S_X, or X! (the last read "X factorial").
• For a subset A of the domain and subset B of the codomain we have:

|f(A)| = |A| and |f^-1(B)| = |B|.

Notice that a one-to-one function is injective, but may fail to be surjective, while a one-to-one correspondence is both injective and surjective. In mathematics, the real line is simply the set of real numbers. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics, the factorial of a natural number n is the product of all positive integers less than and equal to n. ...

Bijections and category theory

Formally, bijections are precisely the isomorphisms in the category Set of sets. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ...

See also

• injective function
• isomorphism
• permutation
• symmetric group
• surjective function