Bijection

A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the second set (the codomain) is mapped to from exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set.

A function is bijective if and only if it is invertible; that is, a function $f:X\to Y$ is bijective if and only if there is a function $g:Y\to X,$ the inverse of $f$ , such that each of the two ways for composing the two functions produces an identity function: $g(f(x))=x$ for each $x$ in $X$ and $f(g(y))=y$ for each $y$ in $Y.$

For example, the multiplication by two defines a bijection from the integers to the even numbers, which has the division by two as its inverse function.

A function is bijective if and only if it is both injective (or one-to-one)—meaning that each element in the codomain is mapped to from at most one element of the domain—and surjective (or onto)—meaning that each element of the codomain is mapped to from at least one element of the domain. The term one-to-one correspondence must not be confused with one-to-one function, which means injective but not necessarily surjective.

The elementary operation of counting establishes a bijection from some finite set to the first natural numbers $(1, 2, 3, ...)$ , up to the number of elements in the counted set. It results that two finite sets have the same number of elements if and only if there exists a bijection between them. More generally, two sets are said to have the same cardinal number if there exists a bijection between them.

A bijective function from a set to itself is also called a permutation,^[1] and the set of all permutations of a set forms its symmetric group.

Some bijections with further properties have received specific names, which include automorphisms, isomorphisms, homeomorphisms, diffeomorphisms, permutation groups, and most geometric transformations. Galois correspondences are bijections between sets of mathematical objects of apparently very different nature.

Examples[edit]

Batting line-up of a baseball or cricket team[edit]

Consider the batting line-up of a baseball or cricket team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). The set X will be the players on the team (of size nine in the case of baseball) and the set Y will be the positions in the batting order (1st, 2nd, 3rd, etc.) The "pairing" is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in the same position in the list.

Seats and students of a classroom[edit]

In a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks them to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. What the instructor observed in order to reach this conclusion was that:

The instructor was able to conclude that there were just as many seats as there were students, without having to count either set.

For any set X, the 1_X: X → X, 1_X(x) = x is bijective.

identity function

The function f: R → R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y − 1)/2 such that f(x) = y. More generally, any over the reals, f: R → R, f(x) = ax + b (where a is non-zero) is a bijection. Each real number y is obtained from (or paired with) the real number x = (y − b)/a.

linear function

The function f: R → (−π/2, π/2), given by f(x) = arctan(x) is bijective, since each real number x is paired with exactly one angle y in the interval (−π/2, π/2) so that tan(y) = x (that is, y = arctan(x)). If the (−π/2, π/2) was made larger to include an integer multiple of π/2, then this function would no longer be onto (surjective), since there is no real number which could be paired with the multiple of π/2 by this arctan function.

codomain

The , g: R → R, 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 onto (surjective). However, if the codomain is restricted to the positive real numbers $\mathbb {R} ^{+}\equiv \left(0,\infty \right)$ , then g would be bijective; its inverse (see below) is the natural logarithm function ln.

exponential function

The function h: R → R⁺, h(x) = x² is not bijective: for instance, h(−1) = h(1) = 1, showing that h is not one-to-one (injective). However, if the is restricted to $\mathbb {R} _{0}^{+}\equiv \left[0,\infty \right)$ , then h would be bijective; its inverse is the positive square root function.

domain

By , given any two sets X and Y, and two injective functions f: X → Y and g: Y → X, there exists a bijective function h: X → Y.

Schröder–Bernstein theorem

Cardinality[edit]

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 definition of "same number of elements" (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets.

A function f: R → R is bijective if and only if its meets every horizontal and vertical line exactly once.

graph

If X is a set, then the bijective functions from X to itself, together with the operation of functional composition (∘), form a , the symmetric group of X, which is denoted variously by S(X), S_X, or X! (X factorial).

group

Bijections preserve of sets: for a subset A of the domain with cardinality |A| and subset B of the codomain with cardinality |B|, one has the following equalities:
|f(A)| = |A| and |f⁻¹(B)| = |B|.

cardinalities

If X and Y are with the same cardinality, and f: X → Y, then the following are equivalent:
f is a bijection.

f is a surjection.

f is an injection.

finite sets

For a finite set S, there is a bijection between the set of possible of the elements and the set of bijections from S to S. That is to say, the number of permutations of elements of S is the same as the number of total orderings of that set—namely, n!.

total orderings

Category theory[edit]

Bijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are not always the isomorphisms for more complex categories. For example, in the category Grp of groups, the morphisms must be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphisms which are bijective homomorphisms.

Generalization to partial functions[edit]

The notion of one-to-one correspondence generalizes to partial functions, where they are called partial bijections, although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the symmetric inverse semigroup.^[4]

Another way of defining the same notion is to say that a partial bijection from A to B is any relation R (which turns out to be a partial function) with the property that R is the graph of a bijection f:A′→B′, where A′ is a subset of A and B′ is a subset of B.^[5]

When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation.^[6] An example is the Möbius transformation simply defined on the complex plane, rather than its completion to the extended complex plane.^[7]

Ax–Grothendieck theorem

Bijection, injection and surjection

Bijective numeration

Bijective proof

Category theory

Multivalued function

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This topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory. Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic may be found in any of these:

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Bijection"

"Bijection". MathWorld.