<s>
In	O
mathematics	O
,	O
a	O
bijection	B-Algorithm
,	O
also	O
known	O
as	O
a	O
bijective	B-Algorithm
function	I-Algorithm
,	O
one-to-one	B-Algorithm
correspondence	I-Algorithm
,	O
or	O
invertible	O
function	O
,	O
is	O
a	O
function	O
between	O
the	O
elements	O
of	O
two	O
sets	O
,	O
where	O
each	O
element	O
of	O
one	O
set	O
is	O
paired	O
with	O
exactly	O
one	O
element	O
of	O
the	O
other	O
set	O
,	O
and	O
each	O
element	O
of	O
the	O
other	O
set	O
is	O
paired	O
with	O
exactly	O
one	O
element	O
of	O
the	O
first	O
set	O
;	O
there	O
are	O
no	O
unpaired	O
elements	O
between	O
the	O
two	O
sets	O
.	O
</s>
<s>
In	O
mathematical	O
terms	O
,	O
a	O
bijective	B-Algorithm
function	I-Algorithm
is	O
a	O
one-to-one	O
(	O
injective	O
)	O
and	O
onto	O
(	O
surjective	B-Algorithm
)	O
mapping	O
of	O
a	O
set	O
X	O
to	O
a	O
set	O
Y	O
.	O
</s>
<s>
The	O
term	O
one-to-one	B-Algorithm
correspondence	I-Algorithm
must	O
not	O
be	O
confused	O
with	O
one-to-one	O
function	O
(	O
an	O
injective	O
function	O
;	O
see	O
figures	O
)	O
.	O
</s>
<s>
A	O
bijection	B-Algorithm
from	O
the	O
set	O
X	O
to	O
the	O
set	O
Y	O
has	O
an	O
inverse	O
function	O
from	O
Y	O
to	O
X	O
.	O
</s>
<s>
If	O
X	O
and	O
Y	O
are	O
finite	O
sets	O
,	O
then	O
the	O
existence	O
of	O
a	O
bijection	B-Algorithm
means	O
they	O
have	O
the	O
same	O
number	O
of	O
elements	O
.	O
</s>
<s>
A	O
bijective	B-Algorithm
function	I-Algorithm
from	O
a	O
set	O
to	O
itself	O
is	O
also	O
called	O
a	O
permutation	B-Algorithm
,	O
and	O
the	O
set	O
of	O
all	O
permutations	B-Algorithm
of	O
a	O
set	O
forms	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Bijective	B-Algorithm
functions	I-Algorithm
are	O
essential	O
to	O
many	O
areas	O
of	O
mathematics	O
including	O
the	O
definitions	O
of	O
isomorphisms	O
,	O
homeomorphisms	O
,	O
diffeomorphisms	O
,	O
permutation	B-Algorithm
groups	I-Algorithm
,	O
and	O
projective	B-Algorithm
maps	I-Algorithm
.	O
</s>
<s>
For	O
a	O
pairing	O
between	O
X	O
and	O
Y	O
(	O
where	O
Y	O
need	O
not	O
be	O
different	O
from	O
X	O
)	O
to	O
be	O
a	O
bijection	B-Algorithm
,	O
four	O
properties	O
must	O
hold	O
:	O
</s>
<s>
Satisfying	O
properties	O
(	O
1	O
)	O
and	O
(	O
2	O
)	O
means	O
that	O
a	O
pairing	O
is	O
a	O
function	O
with	O
domain	B-Algorithm
X	O
.	O
</s>
<s>
Functions	O
which	O
satisfy	O
property	O
(	O
3	O
)	O
are	O
said	O
to	O
be	O
"	O
onto	O
Y	O
"	O
and	O
are	O
called	O
surjections	B-Algorithm
(	O
or	O
surjective	B-Algorithm
functions	I-Algorithm
)	O
.	O
</s>
<s>
With	O
this	O
terminology	O
,	O
a	O
bijection	B-Algorithm
is	O
a	O
function	O
which	O
is	O
both	O
a	O
surjection	B-Algorithm
and	O
an	O
injection	O
,	O
or	O
using	O
other	O
words	O
,	O
a	O
bijection	B-Algorithm
is	O
a	O
function	O
which	O
is	O
both	O
"	O
one-to-one	O
"	O
and	O
"	O
onto	O
"	O
.	O
</s>
<s>
Bijections	B-Algorithm
are	O
sometimes	O
denoted	O
by	O
a	O
two-headed	O
rightwards	O
arrow	O
with	O
tail	O
(	O
)	O
,	O
as	O
in	O
f	O
:	O
X	O
⤖	O
Y	O
.	O
</s>
<s>
This	O
symbol	O
is	O
a	O
combination	O
of	O
the	O
two-headed	O
rightwards	O
arrow	O
(	O
)	O
,	O
sometimes	O
used	O
to	O
denote	O
surjections	B-Algorithm
,	O
and	O
the	O
rightwards	O
arrow	O
with	O
a	O
barbed	O
tail	O
(	O
)	O
,	O
sometimes	O
used	O
to	O
denote	O
injections	O
.	O
</s>
<s>
Consider	O
the	O
batting	O
line-up	O
of	O
a	O
baseball	O
or	O
cricket	B-Application
team	O
(	O
or	O
any	O
list	O
of	O
all	O
the	O
players	O
of	O
any	O
sports	O
team	O
where	O
every	O
player	O
holds	O
a	O
specific	O
spot	O
in	O
a	O
line-up	O
)	O
.	O
</s>
<s>
After	O
a	O
quick	O
look	O
around	O
the	O
room	O
,	O
the	O
instructor	O
declares	O
that	O
there	O
is	O
a	O
bijection	B-Algorithm
between	O
the	O
set	O
of	O
students	O
and	O
the	O
set	O
of	O
seats	O
,	O
where	O
each	O
student	O
is	O
paired	O
with	O
the	O
seat	O
they	O
are	O
sitting	O
in	O
.	O
</s>
<s>
For	O
any	O
set	O
X	O
,	O
the	O
identity	O
function	O
1X	O
:	O
X	O
→	O
X	O
,	O
1X(x )	O
=	O
x	O
is	O
bijective	B-Algorithm
.	O
</s>
<s>
The	O
function	O
f	O
:	O
R	O
→	O
R	O
,	O
f(x )	O
=	O
2x	O
+	O
1	O
is	O
bijective	B-Algorithm
,	O
since	O
for	O
each	O
y	O
there	O
is	O
a	O
unique	O
x	O
=	O
(	O
y	O
−	O
1	O
)	O
/2	O
such	O
that	O
f(x )	O
=	O
y	O
.	O
</s>
<s>
More	O
generally	O
,	O
any	O
linear	O
function	O
over	O
the	O
reals	O
,	O
f	O
:	O
R	O
→	O
R	O
,	O
f(x )	O
=	O
ax	O
+	O
b	O
(	O
where	O
a	O
is	O
non-zero	O
)	O
is	O
a	O
bijection	B-Algorithm
.	O
</s>
<s>
The	O
function	O
f	O
:	O
R	O
→	O
(−π/	O
2	O
,	O
π/2	O
)	O
,	O
given	O
by	O
f(x )	O
=	O
arctan(x )	O
is	O
bijective	B-Algorithm
,	O
since	O
each	O
real	O
number	O
x	O
is	O
paired	O
with	O
exactly	O
one	O
angle	O
y	O
in	O
the	O
interval	O
(−π/	O
2	O
,	O
π/2	O
)	O
so	O
that	O
tan(y )	O
=	O
x	O
(	O
that	O
is	O
,	O
y	O
=	O
arctan(x )	O
)	O
.	O
</s>
<s>
If	O
the	O
codomain	B-Algorithm
(−π/	O
2	O
,	O
π/2	O
)	O
was	O
made	O
larger	O
to	O
include	O
an	O
integer	O
multiple	O
of	O
π/2	O
,	O
then	O
this	O
function	O
would	O
no	O
longer	O
be	O
onto	O
(	O
surjective	B-Algorithm
)	O
,	O
since	O
there	O
is	O
no	O
real	O
number	O
which	O
could	O
be	O
paired	O
with	O
the	O
multiple	O
of	O
π/2	O
by	O
this	O
arctan	O
function	O
.	O
</s>
<s>
The	O
exponential	O
function	O
,	O
g	O
:	O
R	O
→	O
R	O
,	O
g(x )	O
=	O
ex	O
,	O
is	O
not	O
bijective	B-Algorithm
:	O
for	O
instance	O
,	O
there	O
is	O
no	O
x	O
in	O
R	O
such	O
that	O
g(x )	O
=	O
−1	O
,	O
showing	O
that	O
g	O
is	O
not	O
onto	O
(	O
surjective	B-Algorithm
)	O
.	O
</s>
<s>
However	O
,	O
if	O
the	O
codomain	B-Algorithm
is	O
restricted	O
to	O
the	O
positive	O
real	O
numbers	O
,	O
then	O
g	O
would	O
be	O
bijective	B-Algorithm
;	O
its	O
inverse	O
(	O
see	O
below	O
)	O
is	O
the	O
natural	O
logarithm	O
function	O
ln	O
.	O
</s>
<s>
The	O
function	O
h	O
:	O
R	O
→	O
R+	O
,	O
h(x )	O
=	O
x2	O
is	O
not	O
bijective	B-Algorithm
:	O
for	O
instance	O
,	O
h( −1	O
)	O
=	O
h(1 )	O
=	O
1	O
,	O
showing	O
that	O
h	O
is	O
not	O
one-to-one	O
(	O
injective	O
)	O
.	O
</s>
<s>
However	O
,	O
if	O
the	O
domain	B-Algorithm
is	O
restricted	O
to	O
,	O
then	O
h	O
would	O
be	O
bijective	B-Algorithm
;	O
its	O
inverse	O
is	O
the	O
positive	O
square	O
root	O
function	O
.	O
</s>
<s>
By	O
Cantor-Bernstein-Schröder	O
theorem	O
,	O
given	O
any	O
two	O
sets	O
X	O
and	O
Y	O
,	O
and	O
two	O
injective	O
functions	O
f	O
:	O
X	O
→	O
Y	O
and	O
g	O
:	O
Y	O
→	O
X	O
,	O
there	O
exists	O
a	O
bijective	B-Algorithm
function	I-Algorithm
h	O
:	O
X	O
→	O
Y	O
.	O
</s>
<s>
A	O
bijection	B-Algorithm
f	O
with	O
domain	B-Algorithm
X	O
(	O
indicated	O
by	O
f	O
:	O
X	O
→	O
Y	O
in	O
functional	O
notation	O
)	O
also	O
defines	O
a	O
converse	O
relation	O
starting	O
in	O
Y	O
and	O
going	O
to	O
X	O
(	O
by	O
turning	O
the	O
arrows	O
around	O
)	O
.	O
</s>
<s>
The	O
process	O
of	O
"	O
turning	O
the	O
arrows	O
around	O
"	O
for	O
an	O
arbitrary	O
function	O
does	O
not	O
,	O
in	O
general	O
,	O
yield	O
a	O
function	O
,	O
but	O
properties	O
(	O
3	O
)	O
and	O
(	O
4	O
)	O
of	O
a	O
bijection	B-Algorithm
say	O
that	O
this	O
inverse	O
relation	O
is	O
a	O
function	O
with	O
domain	B-Algorithm
Y	O
.	O
</s>
<s>
Moreover	O
,	O
properties	O
(	O
1	O
)	O
and	O
(	O
2	O
)	O
then	O
say	O
that	O
this	O
inverse	O
function	O
is	O
a	O
surjection	B-Algorithm
and	O
an	O
injection	O
,	O
that	O
is	O
,	O
the	O
inverse	O
function	O
exists	O
and	O
is	O
also	O
a	O
bijection	B-Algorithm
.	O
</s>
<s>
A	O
function	O
is	O
invertible	O
if	O
and	O
only	O
if	O
it	O
is	O
a	O
bijection	B-Algorithm
.	O
</s>
<s>
Since	O
this	O
function	O
is	O
a	O
bijection	B-Algorithm
,	O
it	O
has	O
an	O
inverse	O
function	O
which	O
takes	O
as	O
input	O
a	O
position	O
in	O
the	O
batting	O
order	O
and	O
outputs	O
the	O
player	O
who	O
will	O
be	O
batting	O
in	O
that	O
position	O
.	O
</s>
<s>
The	O
composition	B-Application
of	O
two	O
bijections	B-Algorithm
f	O
:	O
X	O
→	O
Y	O
and	O
g	O
:	O
Y	O
→	O
Z	O
is	O
a	O
bijection	B-Algorithm
,	O
whose	O
inverse	O
is	O
given	O
by	O
is	O
.	O
</s>
<s>
Conversely	O
,	O
if	O
the	O
composition	B-Application
of	O
two	O
functions	O
is	O
bijective	B-Algorithm
,	O
it	O
only	O
follows	O
that	O
f	O
is	O
injective	O
and	O
g	O
is	O
surjective	B-Algorithm
.	O
</s>
<s>
If	O
X	O
and	O
Y	O
are	O
finite	O
sets	O
,	O
then	O
there	O
exists	O
a	O
bijection	B-Algorithm
between	O
the	O
two	O
sets	O
X	O
and	O
Y	O
if	O
and	O
only	O
if	O
X	O
and	O
Y	O
have	O
the	O
same	O
number	O
of	O
elements	O
.	O
</s>
<s>
A	O
function	O
f	O
:	O
R	O
→	O
R	O
is	O
bijective	B-Algorithm
if	O
and	O
only	O
if	O
its	O
graph	B-Application
meets	O
every	O
horizontal	O
and	O
vertical	O
line	O
exactly	O
once	O
.	O
</s>
<s>
If	O
X	O
is	O
a	O
set	O
,	O
then	O
the	O
bijective	B-Algorithm
functions	I-Algorithm
from	O
X	O
to	O
itself	O
,	O
together	O
with	O
the	O
operation	O
of	O
functional	B-Application
composition	I-Application
( ∘	O
)	O
,	O
form	O
a	O
group	O
,	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
X	O
,	O
which	O
is	O
denoted	O
variously	O
by	O
S(X )	O
,	O
SX	O
,	O
or	O
X	O
!	O
</s>
<s>
Bijections	B-Algorithm
preserve	O
cardinalities	B-Application
of	O
sets	O
:	O
for	O
a	O
subset	O
A	O
of	O
the	O
domain	B-Algorithm
with	O
cardinality	B-Application
|A|	O
and	O
subset	O
B	O
of	O
the	O
codomain	B-Algorithm
with	O
cardinality	B-Application
|B|	O
,	O
one	O
has	O
the	O
following	O
equalities	O
:	O
</s>
<s>
If	O
X	O
and	O
Y	O
are	O
finite	O
sets	O
with	O
the	O
same	O
cardinality	B-Application
,	O
and	O
f	O
:	O
X	O
→	O
Y	O
,	O
then	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
f	O
is	O
a	O
bijection	B-Algorithm
.	O
</s>
<s>
f	O
is	O
a	O
surjection	B-Algorithm
.	O
</s>
<s>
For	O
a	O
finite	O
set	O
S	O
,	O
there	O
is	O
a	O
bijection	B-Algorithm
between	O
the	O
set	O
of	O
possible	O
total	O
orderings	O
of	O
the	O
elements	O
and	O
the	O
set	O
of	O
bijections	B-Algorithm
from	O
S	O
to	O
S	O
.	O
That	O
is	O
to	O
say	O
,	O
the	O
number	O
of	O
permutations	B-Algorithm
of	O
elements	O
of	O
S	O
is	O
the	O
same	O
as	O
the	O
number	O
of	O
total	O
orderings	O
of	O
that	O
set	O
—	O
namely	O
,	O
n	O
!	O
.	O
</s>
<s>
Bijections	B-Algorithm
are	O
precisely	O
the	O
isomorphisms	O
in	O
the	O
category	O
Set	O
of	O
sets	O
and	O
set	O
functions	O
.	O
</s>
<s>
However	O
,	O
the	O
bijections	B-Algorithm
are	O
not	O
always	O
the	O
isomorphisms	O
for	O
more	O
complex	O
categories	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
the	O
category	O
Grp	O
of	O
groups	O
,	O
the	O
morphisms	O
must	O
be	O
homomorphisms	O
since	O
they	O
must	O
preserve	O
the	O
group	O
structure	O
,	O
so	O
the	O
isomorphisms	O
are	O
group	O
isomorphisms	O
which	O
are	O
bijective	B-Algorithm
homomorphisms	O
.	O
</s>
<s>
The	O
notion	O
of	O
one-to-one	B-Algorithm
correspondence	I-Algorithm
generalizes	O
to	O
partial	B-Algorithm
functions	I-Algorithm
,	O
where	O
they	O
are	O
called	O
partial	O
bijections	B-Algorithm
,	O
although	O
partial	O
bijections	B-Algorithm
are	O
only	O
required	O
to	O
be	O
injective	O
.	O
</s>
<s>
The	O
reason	O
for	O
this	O
relaxation	O
is	O
that	O
a	O
(	O
proper	O
)	O
partial	B-Algorithm
function	I-Algorithm
is	O
already	O
undefined	O
for	O
a	O
portion	O
of	O
its	O
domain	B-Algorithm
;	O
thus	O
there	O
is	O
no	O
compelling	O
reason	O
to	O
constrain	O
its	O
inverse	O
to	O
be	O
a	O
total	B-Algorithm
function	I-Algorithm
,	O
i.e.	O
</s>
<s>
defined	O
everywhere	O
on	O
its	O
domain	B-Algorithm
.	O
</s>
<s>
The	O
set	O
of	O
all	O
partial	O
bijections	B-Algorithm
on	O
a	O
given	O
base	O
set	O
is	O
called	O
the	O
symmetric	O
inverse	O
semigroup	O
.	O
</s>
<s>
R	O
(	O
which	O
turns	O
out	O
to	O
be	O
a	O
partial	B-Algorithm
function	I-Algorithm
)	O
with	O
the	O
property	O
that	O
R	O
is	O
the	O
graph	B-Application
of	I-Application
a	O
bijection	B-Algorithm
f:A	O
′→B′	O
,	O
where	O
A′	O
is	O
a	O
subset	O
of	O
A	O
and	O
B′	O
is	O
a	O
subset	O
of	O
B	O
.	O
</s>
<s>
When	O
the	O
partial	O
bijection	B-Algorithm
is	O
on	O
the	O
same	O
set	O
,	O
it	O
is	O
sometimes	O
called	O
a	O
one-to-one	O
partial	O
transformation	B-Algorithm
.	O
</s>
<s>
An	O
example	O
is	O
the	O
Möbius	O
transformation	B-Algorithm
simply	O
defined	O
on	O
the	O
complex	O
plane	O
,	O
rather	O
than	O
its	O
completion	O
to	O
the	O
extended	O
complex	O
plane	O
.	O
</s>
