Write a note on functional dependencies with examples?

Answer 1

Some P

oints to Note on Functional Dependencies, Minimal Co

ve

rs, and 3NF/BCNF decompositions

•T

ri

vial functional dependenc

yv

s. non-tri

vial functional dependenc

y

Af

unctional dependenc

yX

→

Yi

st

ri

vial if Y

,t

he right hand side of the fd,

is a

subset of X.

Fo

re

xample, the functional dependenc

y"

(S#, P#)

→

S#" is tri

vial because the set

{S#} (for the R.H.S. of the fd) is a subset of (S#, P#} (for the L.H.S. of the fd).

On the other hand, the functional dependenc

y"

(S#, P#)

→

S#, Qty" is NON-tri

vial

because the set {S#, Qty} is NO

Tas

ubset of the attrib

ute set {S#, P#}.

•C

an we remo

ve a

ttrib

utes common to the L.H.S. and R.H.S. of a functional depen-

denc

y?

NO, in general, we can NO

Td

ot

hat - namely

,i

fw

eh

av e

"(X, Y)

→

X, Z", we can

NO

Ts

imply remo

ve t

he attrib

ute "X" common to both the L.H.S. and R.H.S. of the

fd, and obtain "Y

→

Z".

Fo

re

xample, we may ha

ve "

(S#, P#)

→

Qty" and thus "(S#, P#)

→

S#, Qty" (by

augmentation rule).

Ho

we

ve

r, f

rom "(S#, P#)

→

S#, Qty", we can NO

Td

eri

ve "

P#

→

Qty".

•W

hich functional dependenc

yt

or

emo

ve

?

Suppose we ha

ve a r

elation R(ABC) with functional dependencies F = {A

→

B, B

→

C, A

→

C}. W

ec

an see that "A

→

C" can be deri

ve

df

rom "A

→

B" and "B

→

C". Here

we should remo

ve "

A

→

C" when we construct minimal co

ve

rf

or F

,

since "A

→

C" can be deri

ve

df

rom the remaining functional dependencies.

If we remo

ve

d"

A

→

B" or "B

→

C" instead of remo

ving "A

→

C", we w

ould not

be able to re-deri

ve t

he remo

ve

do

ne from the remaining fd'

s.

So the idea is to remo

ve t

he fd'

sw

hich are deri

va

ble from the others, and k

eep

those fd'

su

sed in the process of deri

va

tion.