Bijection

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In mathematics, a function is a bijection, or is bijective, if it is both injective and surjective. A bijection

f

from a set

X

to a set

Y

determines a one-to-one correspondence between the sets

X

and

Y

; for every

x \in X

there is a unique

y\in Y

with

f(x) = y

, and for every

y \in Y

there is a unique

x\in X

with

y=f(x)

The example to the right shows a bijection between the sets $\{1,2,3,4\}$ and $\{A,B,C,D\}$ .

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.

A bijective function from a set to itself is also called a ''permutation''.

Composition and inverses

A function

f:X \rightarrow Y

is bijective if and only if it satisfies the condition :for every

y \in Y

there is a unique

x\in X

with

y=f(x)

This is precisely the condition for the relation $y=f(x)$ to define a function from $Y$ to $X$ , so a function is a bijection if and only if it is invertible. In particular, the inverse will also be a bijection.

The composition $\scriptstyle g \,\circ\, f$ of two bijections $\scriptstyle f:\,X \,\to\, Y$ and $\scriptstyle g:\, Y \,\to\, Z$ is a bijection. The inverse of $\scriptstyle g \,\circ\, f$ is $\scriptstyle (g \,\circ\, f)^{-1} \;=\; (f^{-1}) \,\circ\, (g^{-1})$ .

Conversely, if the composition $\scriptstyle g \,\circ\, f$ of two functions is bijective, we can only say that ''f'' is injective and ''g'' is surjective.

Bijections and cardinality

If ''X'' and ''Y'' are finite sets, then there exists a bijection between the two sets ''X'' and ''Y'' iff ''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", 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 non-examples

For any set ''X'', the identity function

\scriptstyle id_X:\; X \,\rightarrow\, X,\; id_X(x) \;=\; x

, is bijective.

The function

\scriptstyle f:\; \R \,\rightarrow\, \R,\; f(x) \;=\; 2x \,+\, 1

is bijective, since for each ''y'' there is a unique

\scriptstyle x \;=\; \frac{1}{2}\left(y \,-\, 1\right)

such that

\scriptstyle f(x) \;=\; y

The exponential function,

\scriptstyle g:\; \R \,\rightarrow\, \R,\; g(x) \;=\; e^x

, is not bijective: for instance, there is no

\scriptstyle x \,\in\, \R \,|\, g(x) \;=\; -1

, showing that ''g'' is not surjective. However if the codomain is restricted to the positive real numbers

\scriptstyle \R^+ \;\equiv\; \left(0,\, +\infty\right)

, then ''g'' becomes bijective; its inverse is the natural logarithm function ln.

The function

\scriptstyle h:\; \R \,\rightarrow\, \R^+,\; g(x) \;=\; x^2

is not bijective: for instance,

\scriptstyle h(-1) \;=\; h(+1) \;=\; 1

, showing that ''h'' is not injective. However, if the domain is restricted to

\scriptstyle\R^+_0 \;\equiv\; \left[0,\, +\infty\right)

, then ''h'' becomes bijective; its inverse is the positive square root function.

Properties

A function

\scriptstyle f:\; \R \,\rightarrow\, \R

is bijective if and only if its graph meets every horizontal and vertical line exactly once.

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

Bijections preserve cardinalities 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''|.

If ''X'' and ''Y'' are finite sets with the same cardinality, and

\scriptstyle f:\; X \,\rightarrow\, Y

, then the following are equivalent:

# ''f'' is a bijection.

# ''f'' is a surjection.

# ''f'' is an injection.

For a finite set ''S'', there is a bijection between the set of possible total orderings 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''!.

Bijections and category theory

Bijections are precisely the isomorphisms in the category Set of sets and functions. However, the bijections are not always the isomorphisms. For example, in the category Top of topological spaces and continuous functions, the isomorphisms must be homeomorphisms in addition to being bijections.

References

External links

Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.

Category:Functions and mappings Category:Basic concepts in set theory Category:Mathematical relations Category:Types of functions

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