<s>
Armstrong	B-Application
's	I-Application
axioms	I-Application
are	O
a	O
set	O
of	O
references	O
(	O
or	O
,	O
more	O
precisely	O
,	O
inference	O
rules	O
)	O
used	O
to	O
infer	O
all	O
the	O
functional	B-Application
dependencies	I-Application
on	O
a	O
relational	B-Application
database	I-Application
.	O
</s>
<s>
The	O
axioms	O
are	O
sound	O
in	O
generating	O
only	O
functional	B-Application
dependencies	I-Application
in	O
the	O
closure	O
of	O
a	O
set	O
of	O
functional	B-Application
dependencies	I-Application
(	O
denoted	O
as	O
)	O
when	O
applied	O
to	O
that	O
set	O
(	O
denoted	O
as	O
)	O
.	O
</s>
<s>
They	O
are	O
also	O
complete	O
in	O
that	O
repeated	O
application	O
of	O
these	O
rules	O
will	O
generate	O
all	O
functional	B-Application
dependencies	I-Application
in	O
the	O
closure	O
.	O
</s>
<s>
More	O
formally	O
,	O
let	O
denote	O
a	O
relational	O
scheme	O
over	O
the	O
set	O
of	O
attributes	O
with	O
a	O
set	O
of	O
functional	B-Application
dependencies	I-Application
.	O
</s>
<s>
We	O
say	O
that	O
a	O
functional	B-Application
dependency	I-Application
is	O
logically	O
implied	O
by	O
,	O
and	O
denote	O
it	O
with	O
if	O
and	O
only	O
if	O
for	O
every	O
instance	O
of	O
that	O
satisfies	O
the	O
functional	B-Application
dependencies	I-Application
in	O
,	O
also	O
satisfies	O
.	O
</s>
<s>
We	O
denote	O
by	O
the	O
set	O
of	O
all	O
functional	B-Application
dependencies	I-Application
that	O
are	O
logically	O
implied	O
by	O
.	O
</s>
<s>
Furthermore	O
,	O
with	O
respect	O
to	O
a	O
set	O
of	O
inference	O
rules	O
,	O
we	O
say	O
that	O
a	O
functional	B-Application
dependency	I-Application
is	O
derivable	O
from	O
the	O
functional	B-Application
dependencies	I-Application
in	O
by	O
the	O
set	O
of	O
inference	O
rules	O
,	O
and	O
we	O
denote	O
it	O
by	O
if	O
and	O
only	O
if	O
is	O
obtainable	O
by	O
means	O
of	O
repeatedly	O
applying	O
the	O
inference	O
rules	O
in	O
to	O
functional	B-Application
dependencies	I-Application
in	O
.	O
</s>
<s>
We	O
denote	O
by	O
the	O
set	O
of	O
all	O
functional	B-Application
dependencies	I-Application
that	O
are	O
derivable	O
from	O
by	O
inference	O
rules	O
in	O
.	O
</s>
<s>
that	O
is	O
to	O
say	O
,	O
we	O
cannot	O
derive	O
by	O
means	O
of	O
functional	B-Application
dependencies	I-Application
that	O
are	O
not	O
logically	O
implied	O
by	O
.	O
</s>
<s>
more	O
simply	O
put	O
,	O
we	O
are	O
able	O
to	O
derive	O
by	O
all	O
the	O
functional	B-Application
dependencies	I-Application
that	O
are	O
logically	O
implied	O
by	O
.	O
</s>
<s>
Henceforth	O
we	O
will	O
denote	O
by	O
letters	O
,	O
,	O
any	O
subset	O
of	O
and	O
,	O
for	O
short	O
,	O
the	O
union	O
of	O
two	O
sets	O
of	O
attributes	O
and	O
by	O
instead	O
of	O
the	O
usual	O
;	O
this	O
notation	O
is	O
rather	O
standard	O
in	O
database	B-General_Concept
theory	I-General_Concept
when	O
dealing	O
with	O
sets	O
of	O
attributes	O
.	O
</s>
<s>
Given	O
a	O
set	O
of	O
functional	B-Application
dependencies	I-Application
,	O
an	O
Armstrong	O
relation	O
is	O
a	O
relation	O
which	O
satisfies	O
all	O
the	O
functional	B-Application
dependencies	I-Application
in	O
the	O
closure	O
and	O
only	O
those	O
dependencies	O
.	O
</s>
