<s>
In	O
database	B-General_Concept
theory	I-General_Concept
,	O
a	O
multivalued	B-Application
dependency	I-Application
is	O
a	O
full	O
constraint	O
between	O
two	O
sets	O
of	O
attributes	O
in	O
a	O
relation	B-Application
.	O
</s>
<s>
In	O
contrast	O
to	O
the	O
functional	B-Application
dependency	I-Application
,	O
the	O
multivalued	B-Application
dependency	I-Application
requires	O
that	O
certain	O
tuples	B-Application
be	O
present	O
in	O
a	O
relation	B-Application
.	O
</s>
<s>
Therefore	O
,	O
a	O
multivalued	B-Application
dependency	I-Application
is	O
a	O
special	O
case	O
of	O
tuple-generating	O
dependency	O
.	O
</s>
<s>
The	O
multivalued	B-Application
dependency	I-Application
plays	O
a	O
role	O
in	O
the	O
4NF	O
database	B-Application
normalization	I-Application
.	O
</s>
<s>
A	O
multivalued	B-Application
dependency	I-Application
is	O
a	O
special	O
case	O
of	O
a	O
join	O
dependency	O
,	O
with	O
only	O
two	O
sets	O
of	O
values	O
involved	O
,	O
i.e.	O
</s>
<s>
A	O
multivalued	B-Application
dependency	I-Application
exists	O
when	O
there	O
are	O
at	O
least	O
three	O
attributes	O
(	O
like	O
X	O
,	O
Y	O
and	O
Z	O
)	O
in	O
a	O
relation	B-Application
and	O
for	O
a	O
value	O
of	O
X	O
there	O
is	O
a	O
well	O
defined	O
set	O
of	O
values	O
of	O
Y	O
and	O
a	O
well	O
defined	O
set	O
of	O
values	O
of	O
Z	O
.	O
</s>
<s>
Let	O
be	O
a	O
relation	B-Application
and	O
let	O
and	O
be	O
sets	O
of	O
attributes	O
.	O
</s>
<s>
The	O
multivalued	B-Application
dependency	I-Application
(	O
"	O
multidetermines	O
"	O
)	O
holds	O
on	O
if	O
,	O
for	O
any	O
legal	O
relation	B-Application
and	O
all	O
pairs	O
of	O
tuples	B-Application
and	O
in	O
such	O
that	O
,	O
there	O
exist	O
tuples	B-Application
and	O
in	O
such	O
that	O
:	O
</s>
<s>
Informally	O
,	O
if	O
one	O
denotes	O
by	O
the	O
tuple	B-Application
having	O
values	O
for	O
collectively	O
equal	O
to	O
,	O
then	O
whenever	O
the	O
tuples	B-Application
and	O
exist	O
in	O
,	O
the	O
tuples	B-Application
and	O
should	O
also	O
exist	O
in	O
.	O
</s>
<s>
The	O
multivalued	B-Application
dependency	I-Application
can	O
be	O
schematically	O
depicted	O
as	O
shown	O
below	O
:	O
</s>
<s>
Consider	O
this	O
example	O
of	O
a	O
relation	B-Application
of	O
university	O
courses	O
,	O
the	O
books	O
recommended	O
for	O
the	O
course	O
,	O
and	O
the	O
lecturers	O
who	O
will	O
be	O
teaching	O
the	O
course	O
:	O
</s>
<s>
Because	O
the	O
lecturers	O
attached	O
to	O
the	O
course	O
and	O
the	O
books	O
attached	O
to	O
the	O
course	O
are	O
independent	O
of	O
each	O
other	O
,	O
this	O
database	O
design	O
has	O
a	O
multivalued	B-Application
dependency	I-Application
;	O
if	O
we	O
were	O
to	O
add	O
a	O
new	O
book	O
to	O
the	O
AHA	O
course	O
,	O
we	O
would	O
have	O
to	O
add	O
one	O
record	O
for	O
each	O
of	O
the	O
lecturers	O
on	O
that	O
course	O
,	O
and	O
vice	O
versa	O
.	O
</s>
<s>
Put	O
formally	O
,	O
there	O
are	O
two	O
multivalued	B-Application
dependencies	I-Application
in	O
this	O
relation	B-Application
:	O
 { course }  { book } 	O
and	O
equivalently	O
 { course }  { lecturer } 	O
.	O
</s>
<s>
Databases	O
with	O
multivalued	B-Application
dependencies	I-Application
thus	O
exhibit	O
redundancy	O
.	O
</s>
<s>
In	O
database	B-Application
normalization	I-Application
,	O
fourth	O
normal	B-Application
form	I-Application
requires	O
that	O
for	O
every	O
nontrivial	O
multivalued	B-Application
dependency	I-Application
XY	O
,	O
X	O
is	O
a	O
superkey	B-Application
.	O
</s>
<s>
A	O
multivalued	B-Application
dependency	I-Application
X	O
Y	O
is	O
trivial	O
if	O
Y	O
is	O
a	O
subset	O
of	O
X	O
,	O
or	O
if	O
is	O
the	O
whole	O
set	O
of	O
attributes	O
of	O
the	O
relation	B-Application
.	O
</s>
<s>
The	O
following	O
also	O
involve	O
functional	B-Application
dependencies	I-Application
:	O
</s>
<s>
A	O
decomposition	O
of	O
R	O
into	O
(	O
X	O
,	O
Y	O
)	O
and	O
(	O
X	O
,	O
RY	O
)	O
is	O
a	O
lossless-join	B-Algorithm
decomposition	I-Algorithm
if	O
and	O
only	O
if	O
XY	O
holds	O
inR	O
.	O
</s>
<s>
Every	O
FD	O
is	O
an	O
MVD	O
because	O
if	O
X	O
Y	O
,	O
then	O
swapping	O
Y	O
's	O
between	O
tuples	B-Application
that	O
agree	O
on	O
X	O
does	O
n't	O
create	O
new	O
tuples	B-Application
.	O
</s>
<s>
Closure	O
of	O
a	O
set	O
of	O
MVDs	O
is	O
the	O
set	O
of	O
all	O
MVDs	O
that	O
can	O
be	O
inferred	O
using	O
the	O
following	O
rules	O
(	O
Armstrong	B-Application
's	I-Application
axioms	I-Application
)	O
:	O
</s>
<s>
That	O
a	O
multivalued	B-Application
dependency	I-Application
is	O
a	O
full	O
constraint	O
follows	O
from	O
its	O
definition	O
,	O
as	O
where	O
it	O
says	O
something	O
about	O
the	O
attributes	O
.	O
</s>
<s>
tuple-generating	O
dependency	O
A	O
dependency	O
which	O
explicitly	O
requires	O
certain	O
tuples	B-Application
to	O
be	O
present	O
in	O
the	O
relation	B-Application
.	O
</s>
<s>
trivial	O
multivalued	B-Application
dependency	I-Application
1	O
A	O
multivalued	B-Application
dependency	I-Application
which	O
involves	O
all	O
the	O
attributes	O
of	O
a	O
relation	B-Application
e	O
..	O
A	O
trivial	O
multivalued	B-Application
dependency	I-Application
implies	O
,	O
for	O
tuples	B-Application
and	O
,	O
tuples	B-Application
and	O
which	O
are	O
equal	O
to	O
and	O
.	O
</s>
<s>
trivial	O
multivalued	B-Application
dependency	I-Application
2	O
A	O
multivalued	B-Application
dependency	I-Application
for	O
which	O
.	O
</s>
