<s>
In	O
mathematics	O
,	O
a	O
finitary	B-Algorithm
relation	I-Algorithm
over	O
sets	O
is	O
a	O
subset	O
of	O
the	O
Cartesian	O
product	O
;	O
that	O
is	O
,	O
it	O
is	O
a	O
set	O
of	O
n-tuples	O
consisting	O
of	O
elements	O
xi	O
in	O
Xi	O
.	O
</s>
<s>
A	O
relation	O
with	O
n	O
"	O
places	O
"	O
is	O
variously	O
called	O
an	O
n-ary	B-Algorithm
relation	I-Algorithm
,	O
an	O
n-adic	O
relation	O
or	O
a	O
relation	O
of	O
degree	O
n	O
.	O
Relations	O
with	O
a	O
finite	O
number	O
of	O
places	O
are	O
called	O
finitary	B-Algorithm
relations	I-Algorithm
(	O
or	O
simply	O
relations	O
if	O
the	O
context	O
is	O
clear	O
)	O
.	O
</s>
<s>
An	O
n-ary	B-Algorithm
relation	I-Algorithm
over	O
sets	O
is	O
an	O
element	O
of	O
the	O
power	O
set	O
of	O
.	O
</s>
<s>
They	O
are	O
sometimes	O
useful	O
for	O
constructing	O
the	O
base	O
case	O
of	O
an	O
induction	B-Algorithm
argument	O
.	O
</s>
<s>
Binary	O
relations	O
are	O
the	O
most	O
commonly	O
studied	O
form	O
of	O
finitary	B-Algorithm
relations	I-Algorithm
.	O
</s>
<s>
The	O
above	O
table	O
is	O
also	O
a	O
simple	O
example	O
of	O
a	O
relational	B-Application
database	I-Application
,	O
a	O
field	O
with	O
theory	O
rooted	O
in	O
relational	B-Algorithm
algebra	I-Algorithm
and	O
applications	O
in	O
data	O
management	O
.	O
</s>
<s>
Computer	B-General_Concept
scientists	I-General_Concept
,	O
logicians	O
,	O
and	O
mathematicians	O
,	O
however	O
,	O
tend	O
to	O
have	O
different	O
conceptions	O
what	O
a	O
general	O
relation	O
is	O
,	O
and	O
what	O
it	O
is	O
consisted	O
of	O
.	O
</s>
<s>
For	O
example	O
,	O
databases	B-Application
are	O
designed	O
to	O
deal	O
with	O
empirical	O
data	O
,	O
which	O
is	O
by	O
definition	O
finite	O
,	O
whereas	O
in	O
mathematics	O
,	O
relations	O
with	O
infinite	O
arity	O
(	O
i.e.	O
,	O
infinitary	O
relation	O
)	O
are	O
also	O
considered	O
.	O
</s>
<s>
Definition	O
1	O
An	O
n-ary	B-Algorithm
relation	I-Algorithm
R	O
over	O
sets	O
is	O
a	O
subset	O
of	O
the	O
Cartesian	O
product	O
.	O
</s>
<s>
Definition	O
2	O
An	O
n-ary	B-Algorithm
relation	I-Algorithm
R	O
over	O
sets	O
is	O
an	O
(	O
)	O
-tuple	O
where	O
G	O
is	O
a	O
subset	O
of	O
the	O
Cartesian	O
product	O
called	O
the	O
graph	O
of	O
R	O
.	O
</s>
<s>
When	O
,	O
where	O
,	O
,	O
,	O
and	O
is	O
a	O
partition	O
of	O
,	O
R	O
is	O
said	O
to	O
be	O
unique	O
on	O
,	O
and	O
is	O
called	O
a	O
primary	B-Application
key	I-Application
of	O
R	O
.	O
In	O
the	O
case	O
where	O
R	O
is	O
a	O
binary	O
relation	O
,	O
when	O
R	O
is	O
unique	O
on	O
{X1},	O
it	O
is	O
also	O
said	O
to	O
be	O
left-unique	O
or	O
injective	O
,	O
and	O
when	O
R	O
is	O
unique	O
on	O
{X2},	O
it	O
is	O
also	O
said	O
to	O
be	O
right-unique	O
or	O
functional	O
.	O
</s>
<s>
When	O
all	O
Xi	O
are	O
the	O
same	O
set	O
X	O
,	O
it	O
is	O
simpler	O
to	O
refer	O
to	O
R	O
as	O
an	O
n-ary	B-Algorithm
relation	I-Algorithm
over	O
X	O
,	O
called	O
a	O
homogeneous	O
relation	O
.	O
</s>
<s>
In	O
applied	O
mathematics	O
,	O
computer	B-General_Concept
science	I-General_Concept
and	O
statistics	O
,	O
it	O
is	O
common	O
to	O
refer	O
to	O
a	O
Boolean-valued	O
function	O
as	O
an	O
n-ary	O
predicate	B-Algorithm
.	O
</s>
<s>
From	O
the	O
more	O
abstract	O
viewpoint	O
of	O
formal	O
logic	O
and	O
model	O
theory	O
,	O
the	O
relation	O
R	O
constitutes	O
a	O
logical	O
model	O
or	O
a	O
relational	O
structure	O
,	O
that	O
serves	O
as	O
one	O
of	O
many	O
possible	O
interpretations	O
of	O
some	O
n-ary	O
predicate	B-Algorithm
symbol	O
.	O
</s>
<s>
He	O
also	O
stated	O
the	O
first	O
formal	O
results	O
in	O
the	O
theory	B-Algorithm
of	I-Algorithm
relations	I-Algorithm
(	O
on	O
De	O
Morgan	O
and	O
relations	O
,	O
see	O
Merrill	O
1990	O
)	O
.	O
</s>
<s>
Charles	O
Peirce	O
,	O
Gottlob	O
Frege	O
,	O
Georg	O
Cantor	O
,	O
Richard	O
Dedekind	O
and	O
others	O
advanced	O
the	O
theory	B-Algorithm
of	I-Algorithm
relations	I-Algorithm
.	O
</s>
<s>
In	O
1970	O
,	O
Edgar	O
Codd	O
proposed	O
a	O
relational	B-Architecture
model	I-Architecture
for	O
databases	B-Application
,	O
thus	O
anticipating	O
the	O
development	O
of	O
data	B-Application
base	I-Application
management	I-Application
systems	I-Application
.	O
</s>
