Bijective

In Mathematics , a bijection, or a '''bijective function''' is a Function ''f'' from a Set ''X'' to a set ''Y'' with the property that, for every ''y'' in ''Y'', there is exactly one ''x'' in ''X'' such that
''f''(''x'') = ''y''.

Alternatively, ''f'' is bijective if it is a one-to-one correspondence between those sets; .)

For example, consider the function succ, defined from the set of Integer s

\Z

\Z

, 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 ''X''

{}\leftrightarrow{}

''Y''.

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 also a bijection.

The Composition ''g'' o ''f'' of two bijections ''f''

\;:\;

''X''

{}\leftrightarrow{}

''Y'' and ''g''

\;:\;

''Y''

{}\leftrightarrow{}

''Z'' is a bijection. The inverse of ''g'' o ''f'' is (''g'' o ''f'')⁻¹ = (''f''⁻¹) o (''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 ''g'' o ''f'' is the Identity Function on ''X'', and ''f'' o ''g'' is the Identity Function on ''Y''. Consequently, the sets have the same cardinality.

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 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'') = 2''x'' + 1 is bijective, since for each ''y'' there is a unique ''x'' = (''y'' − 1)/2 such that ''f''(''x'') = ''y''.

The function ln.

The function ''h'' : R $ightarrow$ [0,+∞) with ''h(x)'' = ''x''&² 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.

$\mathbb{R} o \mathbb{R} : x \mapsto (x-1)x(x+1) = x^3 - x$ is not a bijection because −1, 0, and +1 are all in the domain and all map to 0.

$\mathbb{R} o$ {Link without Title} : x \mapsto \sin(x) is not a bijection because π/3 and 2π/3 are both in the domain and both map to (√3)/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 reads "''X'' Factorial ").

For a subset ''A'' of the domain and subset ''B'' of the codomain we have:

	CATEGORIES ABOUT BIJECTION

	functions and mappings

	basic concepts in set theory

	mathematical relations

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Bijective