<s>
A	O
relation	B-Algorithm
of	I-Algorithm
degree	I-Algorithm
zero	I-Algorithm
,	O
0-ary	O
relation	B-Language
,	O
or	O
nullary	O
relation	B-Language
is	O
a	O
relation	B-Language
with	O
zero	O
attributes	O
.	O
</s>
<s>
One	O
has	O
cardinality	B-Application
zero	O
;	O
that	O
is	O
,	O
contains	O
no	O
tuples	B-Application
at	O
all	O
.	O
</s>
<s>
The	O
other	O
has	O
cardinality	B-Application
1	O
contains	O
the	O
unique	O
0-tuple	O
.	O
</s>
<s>
The	O
zero-degree	O
relations	O
represent	O
true	O
and	O
false	O
in	O
relational	B-Algorithm
algebra	I-Algorithm
.	O
</s>
<s>
:57	O
Under	O
the	O
closed-world	B-Application
assumption	I-Application
,	O
an	O
n-ary	O
relation	B-Language
is	O
interpreted	O
as	O
the	O
extension	O
of	O
some	O
n-adic	O
predicate	B-Algorithm
:	O
all	O
and	O
only	O
those	O
n-tuples	O
whose	O
values	O
,	O
substituted	O
for	O
corresponding	O
free	O
variables	O
in	O
the	O
predicate	B-Algorithm
,	O
yield	O
propositions	O
that	O
hold	O
true	O
,	O
appear	O
in	O
the	O
relation	B-Language
.	O
</s>
<s>
A	O
zero-degree	O
relation	B-Language
is	O
therefore	O
interpreted	O
as	O
the	O
extension	O
of	O
the	O
0-adic	O
predicate	B-Algorithm
.	O
</s>
<s>
The	O
zero-degree	O
relation	B-Language
with	O
cardinality	B-Application
zero	O
therefore	O
represents	O
false	O
because	O
it	O
contains	O
no	O
tuples	B-Application
that	O
yield	O
a	O
true	O
proposition	O
,	O
and	O
the	O
zero-degree	O
relation	B-Language
with	O
cardinality	B-Application
1	O
represents	O
true	O
because	O
it	O
contains	O
the	O
unique	O
0-tuple	O
that	O
yields	O
a	O
true	O
proposition	O
.	O
</s>
<s>
The	O
zero-degree	O
relations	O
are	O
also	O
significant	O
as	O
identities	O
for	O
certain	O
operators	O
in	O
the	O
relational	B-Algorithm
algebra	I-Algorithm
.	O
</s>
<s>
The	O
zero-degree	O
relation	B-Language
of	O
cardinality	B-Application
1	O
is	O
the	O
identity	O
with	O
respect	O
to	O
join	O
( ⋈	O
)	O
;	O
that	O
is	O
,	O
when	O
it	O
is	O
joined	O
with	O
any	O
other	O
relation	B-Language
,	O
the	O
result	O
is	O
.	O
</s>
<s>
Since	O
the	O
relational	O
Cartesian	O
product	O
is	O
a	O
special	O
case	O
of	O
join	O
,	O
the	O
zero-degree	O
relation	B-Language
of	O
cardinality	B-Application
1	O
is	O
also	O
the	O
identity	O
with	O
respect	O
to	O
the	O
Cartesian	O
product	O
.	O
</s>
<s>
A	O
projection	B-Algorithm
of	O
a	O
relation	B-Language
over	O
no	O
attributes	O
yields	O
one	O
of	O
the	O
relations	O
of	O
degree	O
zero	O
.	O
</s>
<s>
If	O
the	O
projected	O
relation	B-Language
has	O
cardinality	B-Application
0	O
,	O
the	O
projection	B-Algorithm
will	O
have	O
cardinality	B-Application
0	O
;	O
if	O
the	O
projected	O
relation	B-Language
has	O
positive	O
cardinality	B-Application
,	O
the	O
result	O
will	O
have	O
cardinality	B-Application
1	O
.	O
</s>
<s>
Hugh	O
Darwen	O
refers	O
to	O
the	O
zero-degree	O
relation	B-Language
with	O
cardinality	B-Application
0	O
as	O
TABLE_DUM	O
and	O
the	O
relation	B-Language
with	O
cardinality	B-Application
1	O
as	O
TABLE_DEE	O
,	O
alluding	O
to	O
the	O
characters	O
Tweedledum	O
and	O
Tweedledee	O
.	O
</s>
