<s>
In	O
mathematics	O
,	O
injections	O
,	O
surjections	B-Algorithm
,	O
and	O
bijections	B-Algorithm
are	O
classes	O
of	O
functions	O
distinguished	O
by	O
the	O
manner	O
in	O
which	O
arguments	O
(	O
input	O
expressions	O
from	O
the	O
domain	B-Algorithm
)	O
and	O
images	O
(	O
output	O
expressions	O
from	O
the	O
codomain	B-Algorithm
)	O
are	O
related	O
or	O
mapped	O
to	O
each	O
other	O
.	O
</s>
<s>
A	O
function	O
maps	B-Algorithm
elements	O
from	O
its	O
domain	B-Algorithm
to	O
elements	O
in	O
its	O
codomain	B-Algorithm
.	O
</s>
<s>
The	O
function	O
is	O
injective	O
,	O
or	O
one-to-one	O
,	O
if	O
each	O
element	O
of	O
the	O
codomain	B-Algorithm
is	O
mapped	O
to	O
by	O
at	O
most	O
one	O
element	O
of	O
the	O
domain	B-Algorithm
,	O
or	O
equivalently	O
,	O
if	O
distinct	O
elements	O
of	O
the	O
domain	B-Algorithm
map	O
to	O
distinct	O
elements	O
in	O
the	O
codomain	B-Algorithm
.	O
</s>
<s>
The	O
function	O
is	O
surjective	B-Algorithm
,	O
or	O
onto	O
,	O
if	O
each	O
element	O
of	O
the	O
codomain	B-Algorithm
is	O
mapped	O
to	O
by	O
at	O
least	O
one	O
element	O
of	O
the	O
domain	B-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
the	O
image	O
and	O
the	O
codomain	B-Algorithm
of	O
the	O
function	O
are	O
equal	O
.	O
</s>
<s>
A	O
surjective	B-Algorithm
function	I-Algorithm
is	O
a	O
surjection	B-Algorithm
.	O
</s>
<s>
The	O
function	O
is	O
bijective	B-Algorithm
(	O
one-to-one	B-Algorithm
and	I-Algorithm
onto	I-Algorithm
,	O
one-to-one	B-Algorithm
correspondence	I-Algorithm
,	O
or	O
invertible	O
)	O
if	O
each	O
element	O
of	O
the	O
codomain	B-Algorithm
is	O
mapped	O
to	O
by	O
exactly	O
one	O
element	O
of	O
the	O
domain	B-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
the	O
function	O
is	O
both	O
injective	O
and	O
surjective	B-Algorithm
.	O
</s>
<s>
A	O
bijective	B-Algorithm
function	I-Algorithm
is	O
also	O
called	O
a	O
bijection	B-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
combining	O
the	O
definitions	O
of	O
injective	O
and	O
surjective	B-Algorithm
,	O
</s>
<s>
An	O
injective	O
function	O
need	O
not	O
be	O
surjective	B-Algorithm
(	O
not	O
all	O
elements	O
of	O
the	O
codomain	B-Algorithm
may	O
be	O
associated	O
with	O
arguments	O
)	O
,	O
and	O
a	O
surjective	B-Algorithm
function	I-Algorithm
need	O
not	O
be	O
injective	O
(	O
some	O
images	O
may	O
be	O
associated	O
with	O
more	O
than	O
one	O
argument	O
)	O
.	O
</s>
<s>
The	O
four	O
possible	O
combinations	O
of	O
injective	O
and	O
surjective	B-Algorithm
features	O
are	O
illustrated	O
in	O
the	O
adjacent	O
diagrams	O
.	O
</s>
<s>
A	O
function	O
is	O
injective	O
(	O
one-to-one	O
)	O
if	O
each	O
possible	O
element	O
of	O
the	O
codomain	B-Algorithm
is	O
mapped	O
to	O
by	O
at	O
most	O
one	O
argument	O
.	O
</s>
<s>
Equivalently	O
,	O
a	O
function	O
is	O
injective	O
if	O
it	O
maps	B-Algorithm
distinct	O
arguments	O
to	O
distinct	O
images	O
.	O
</s>
<s>
Since	O
every	O
function	O
is	O
surjective	B-Algorithm
when	O
its	O
codomain	B-Algorithm
is	O
restricted	O
to	O
its	O
image	O
,	O
every	O
injection	O
induces	O
a	O
bijection	B-Algorithm
onto	O
its	O
image	O
.	O
</s>
<s>
More	O
precisely	O
,	O
every	O
injection	O
can	O
be	O
factored	O
as	O
a	O
bijection	B-Algorithm
followed	O
by	O
an	O
inclusion	O
as	O
follows	O
.	O
</s>
<s>
Let	O
be	O
with	O
codomain	B-Algorithm
restricted	O
to	O
its	O
image	O
,	O
and	O
let	O
be	O
the	O
inclusion	O
map	O
from	O
into	O
.	O
</s>
<s>
A	O
dual	O
factorization	O
is	O
given	O
for	O
surjections	B-Algorithm
below	O
.	O
</s>
<s>
The	O
composition	B-Application
of	O
two	O
injections	O
is	O
again	O
an	O
injection	O
,	O
but	O
if	O
is	O
injective	O
,	O
then	O
it	O
can	O
only	O
be	O
concluded	O
that	O
is	O
injective	O
(	O
see	O
figure	O
)	O
.	O
</s>
<s>
A	O
function	O
is	O
surjective	B-Algorithm
or	O
onto	O
if	O
each	O
element	O
of	O
the	O
codomain	B-Algorithm
is	O
mapped	O
to	O
by	O
at	O
least	O
one	O
element	O
of	O
the	O
domain	B-Algorithm
.	O
</s>
<s>
In	O
other	O
words	O
,	O
each	O
element	O
of	O
the	O
codomain	B-Algorithm
has	O
non-empty	O
preimage	O
.	O
</s>
<s>
Equivalently	O
,	O
a	O
function	O
is	O
surjective	B-Algorithm
if	O
its	O
image	O
is	O
equal	O
to	O
its	O
codomain	B-Algorithm
.	O
</s>
<s>
A	O
surjective	B-Algorithm
function	I-Algorithm
is	O
a	O
surjection	B-Algorithm
.	O
</s>
<s>
The	O
following	O
are	O
some	O
facts	O
related	O
to	O
surjections	B-Algorithm
:	O
</s>
<s>
A	O
function	O
is	O
surjective	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
right-invertible	O
,	O
that	O
is	O
,	O
if	O
and	O
only	O
if	O
there	O
is	O
a	O
function	O
such	O
that	O
identity	O
function	O
on	O
.	O
</s>
<s>
By	O
collapsing	O
all	O
arguments	O
mapping	O
to	O
a	O
given	O
fixed	O
image	O
,	O
every	O
surjection	B-Algorithm
induces	O
a	O
bijection	B-Algorithm
from	O
a	O
quotient	O
set	O
of	O
its	O
domain	B-Algorithm
to	O
its	O
codomain	B-Algorithm
.	O
</s>
<s>
More	O
precisely	O
,	O
the	O
preimages	O
under	O
f	O
of	O
the	O
elements	O
of	O
the	O
image	O
of	O
are	O
the	O
equivalence	O
classes	O
of	O
an	O
equivalence	O
relation	O
on	O
the	O
domain	B-Algorithm
of	O
,	O
such	O
that	O
x	O
and	O
y	O
are	O
equivalent	O
if	O
and	O
only	O
they	O
have	O
the	O
same	O
image	O
under	O
.	O
</s>
<s>
As	O
all	O
elements	O
of	O
any	O
one	O
of	O
these	O
equivalence	O
classes	O
are	O
mapped	O
by	O
on	O
the	O
same	O
element	O
of	O
the	O
codomain	B-Algorithm
,	O
this	O
induces	O
a	O
bijection	B-Algorithm
between	O
the	O
quotient	O
set	O
by	O
this	O
equivalence	O
relation	O
(	O
the	O
set	O
of	O
the	O
equivalence	O
classes	O
)	O
and	O
the	O
image	O
of	O
(	O
which	O
is	O
its	O
codomain	B-Algorithm
when	O
is	O
surjective	B-Algorithm
)	O
.	O
</s>
<s>
Moreover	O
,	O
f	O
is	O
the	O
composition	B-Application
of	O
the	O
canonical	O
projection	O
from	O
f	O
to	O
the	O
quotient	O
set	O
,	O
and	O
the	O
bijection	B-Algorithm
between	O
the	O
quotient	O
set	O
and	O
the	O
codomain	B-Algorithm
of	O
.	O
</s>
<s>
The	O
composition	B-Application
of	O
two	O
surjections	B-Algorithm
is	O
again	O
a	O
surjection	B-Algorithm
,	O
but	O
if	O
is	O
surjective	B-Algorithm
,	O
then	O
it	O
can	O
only	O
be	O
concluded	O
that	O
is	O
surjective	B-Algorithm
(	O
see	O
figure	O
)	O
.	O
</s>
<s>
A	O
function	O
is	O
bijective	B-Algorithm
if	O
it	O
is	O
both	O
injective	O
and	O
surjective	B-Algorithm
.	O
</s>
<s>
A	O
bijective	B-Algorithm
function	I-Algorithm
is	O
also	O
called	O
a	O
bijection	B-Algorithm
or	O
a	O
one-to-one	B-Algorithm
correspondence	I-Algorithm
.	O
</s>
<s>
A	O
function	O
is	O
bijective	B-Algorithm
if	O
and	O
only	O
if	O
every	O
possible	O
image	O
is	O
mapped	O
to	O
by	O
exactly	O
one	O
argument	O
.	O
</s>
<s>
The	O
following	O
are	O
some	O
facts	O
related	O
to	O
bijections	B-Algorithm
:	O
</s>
<s>
A	O
function	O
is	O
bijective	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
invertible	O
,	O
that	O
is	O
,	O
there	O
is	O
a	O
function	O
such	O
that	O
identity	O
function	O
on	O
X	O
and	O
identity	O
function	O
on	O
.	O
</s>
<s>
This	O
function	O
maps	B-Algorithm
each	O
image	O
to	O
its	O
unique	O
preimage	O
.	O
</s>
<s>
The	O
composition	B-Application
of	O
two	O
bijections	B-Algorithm
is	O
again	O
a	O
bijection	B-Algorithm
,	O
but	O
if	O
is	O
a	O
bijection	B-Algorithm
,	O
then	O
it	O
can	O
only	O
be	O
concluded	O
that	O
is	O
injective	O
and	O
is	O
surjective	B-Algorithm
(	O
see	O
the	O
figure	O
at	O
right	O
and	O
the	O
remarks	O
above	O
regarding	O
injections	O
and	O
surjections	B-Algorithm
)	O
.	O
</s>
<s>
The	O
bijections	B-Algorithm
from	O
a	O
set	O
to	O
itself	O
form	O
a	O
group	O
under	O
composition	B-Application
,	O
called	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Accordingly	O
,	O
one	O
can	O
define	O
two	O
sets	O
to	O
"	O
have	O
the	O
same	O
number	O
of	O
elements	O
"	O
—	O
if	O
there	O
is	O
a	O
bijection	B-Algorithm
between	O
them	O
.	O
</s>
<s>
In	O
which	O
case	O
,	O
the	O
two	O
sets	O
are	O
said	O
to	O
have	O
the	O
same	O
cardinality	B-Application
.	O
</s>
<s>
Likewise	O
,	O
one	O
can	O
say	O
that	O
set	O
"	O
has	O
fewer	O
than	O
or	O
the	O
same	O
number	O
of	O
elements	O
"	O
as	O
set	O
,	O
if	O
there	O
is	O
an	O
injection	O
from	O
to	O
;	O
one	O
can	O
also	O
say	O
that	O
set	O
"	O
has	O
fewer	O
than	O
the	O
number	O
of	O
elements	O
"	O
in	O
set	O
,	O
if	O
there	O
is	O
an	O
injection	O
from	O
to	O
,	O
but	O
not	O
a	O
bijection	B-Algorithm
between	O
and	O
.	O
</s>
<s>
It	O
is	O
important	O
to	O
specify	O
the	O
domain	B-Algorithm
and	O
codomain	B-Algorithm
of	O
each	O
function	O
,	O
since	O
by	O
changing	O
these	O
,	O
functions	O
which	O
appear	O
to	O
be	O
the	O
same	O
may	O
have	O
different	O
properties	O
.	O
</s>
<s>
For	O
every	O
function	O
,	O
subset	O
of	O
the	O
domain	B-Algorithm
and	O
subset	O
of	O
the	O
codomain	B-Algorithm
,	O
and	O
.	O
</s>
<s>
If	O
is	O
injective	O
,	O
then	O
,	O
and	O
if	O
is	O
surjective	B-Algorithm
,	O
then	O
.	O
</s>
<s>
For	O
every	O
function	O
,	O
one	O
can	O
define	O
a	O
surjection	B-Algorithm
and	O
an	O
injection	O
.	O
</s>
<s>
In	O
the	O
category	O
of	O
sets	O
,	O
injections	O
,	O
surjections	B-Algorithm
,	O
and	O
bijections	B-Algorithm
correspond	O
precisely	O
to	O
monomorphisms	B-Application
,	O
epimorphisms	O
,	O
and	O
isomorphisms	O
,	O
respectively	O
.	O
</s>
<s>
The	B-Application
Oxford	I-Application
English	I-Application
Dictionary	I-Application
records	O
the	O
use	O
of	O
the	O
word	O
injection	O
as	O
a	O
noun	O
by	O
S	O
.	O
Mac	O
Lane	O
in	O
Bulletin	O
of	O
the	O
American	O
Mathematical	O
Society	O
(	O
1950	O
)	O
,	O
and	O
injective	O
as	O
an	O
adjective	O
by	O
Eilenberg	O
and	O
Steenrod	O
in	O
Foundations	O
of	O
Algebraic	O
Topology	O
(	O
1952	O
)	O
.	O
</s>
<s>
However	O
,	O
it	O
was	O
not	O
until	O
the	O
French	O
Bourbaki	O
group	O
coined	O
the	O
injective-surjective-bijective	O
terminology	O
(	O
both	O
as	O
nouns	O
and	O
adjectives	O
)	O
that	O
they	O
achieved	O
widespread	O
adoption	O
.	O
</s>
