Functional dependencies in Database Management System

Topic : FUNCTIONAL
DEPENDENCIES
Prepared By:
Kevin Jadiya
Subject : Database Management System

Functional dependency:
• A dependency occurs in a database when information stored in
the same database table uniquely determines other information
stored in the same table.
or
• You can also describe this as a relationship where knowing the
value of one attribute is enough to tell you the value of another
attribute in the same table.

Definition:
Functional dependency is a relationship that exists
when one attribute uniquely determines another attribute.
• It is represented by an arrow () sign.
•Example:
Acc_no Cus_name Balance address
A1 Rahul 60000 veraval
A2 Meet 78000 Mehsana
A3 Prashant 65000 surat
•if we know the value of account no, we can obtain cus_name,address,
balance etc.
•By this, we can say that address,cus_name and balance is functionally
dependent on account no.

Set of rules to define functional dependency
1. Reflexivity rule:
if Y ⊆ X , then X Y
x  y
Determinant Dependent
Read as…..
x holds y (from left to right)
or
y is dependent on x (from right to left)
Basic terminolog

2. Augmentation rule :
if x  y, then xz  yz for any z.
where , z is a set of attributes.
3. Transitivity rule :
if x  y and y  z ,then x  z.
4. Union rule :
if x  y and x  z, then x  yz.
5. Decomposition rule :
if x  yz,then x  y and x  z.

6. Pseudotransitivity rule :
if x  y and wy  z,then xw  z.
7. Composition rule :
if x  y and w  z then x,w  y,z.
8. Self-determination rule :
x  x

Types of functional dependencies:
1. Full functional dependency :
In a relation, the attribute B is
fully functionally dependent on A, if B is functionally dependent
on A but not on any proper subset of A.
• If we remove one of the attribute from the given relation then
dependency does not exist in fully functional dependency.
•Example :
{roll_no, dept_name}  SPI

2.Partial functional dependency :
In a relationship , the attribute B is
partially functionally dependent on A,if B is functionally
dependent on A as well as on any proper subset of A.
• if there is some attributes that can be remove from A and
then still the dependency holds it is known as partial
dependency.
•Example:
{enrollment_no,dept_name} 
SPI

3. Transitive functional dependency:
in a relation, if attribute
A  B and B  C then C is transitively dependent on A via B.
(provided that A is not functionally dependent on B and C)
•Example:
1. if , {staff_id}  {staff_name} &
{staff_name}  {staff_address}
then , {staff_id}  {staff_address}
2. if , {Book}  {Author}
{Author}  {Author_nationality}
then, {book}  {Author_nationality}

4. Trivial functional dependency:
In a relationship, if attribute
Y is a subset of attribute X then the dependency is known as
trivial functional dependency.
•Example:
x  y where y ⊆ x
{roll_no, dept_name} roll_no

5. Non-trivial functional dependency:
in a relationship, if attribute Y
is not a subset of attribute X then it is known as non-trivial
dependency.
• Example:
x  y where y ⊈ x
{roll_no, dept_name}  student_name

Redundant functional dependency:
A functional dependency
In the set is said to be redundant if it can be derived from the
other functional dependency in the set.

Algorithm to find redundant functional dependency:
Input:
let F be a set of FDs for relation R.
let f: AB is a FD to be examined for redundancy.
Step 1 : F’= F - f # find out new set of FDs by
removing f from F.
Step 2 : T = A # set T equals to determinant of
A  B

Step 3 : for each FD X  Y in F’ do
if….. X ⊆ T then, # If X is contained in T
T=T ⋃ Y # Add Y to T.
end if
Step 4 : if…… B ⊆ T then, # If B is contained in T
f:AB is redundant # Given functional dependency.
end if.
Output :
decision whether given functional dependency is redundan
or not.

Example : suppose a relation R is given with attributes A,B,C,D
& E also set of FDs is given as follow:
F={A  B, C  D, BD  E, AC  E}
(I) Find out whether functional dependency f:AC  E is redundant
or not?
(II) Find out whether functional dependency f:BD E is redundant
or not?
Solution (I): (for f:AC E).....
step 1: F’={AB,CD,BDE} # F’=F-f
step 2: T=AC # Set T= determinant of
AC E.

Step 3: T=AC+B = ACB # AB is in F’ and A ⊆ T.
T=ACB+D = ACBD # C D is in F’ and C ⊆ T.
T=ACBD+E = ACBDE # BD E is in F’ and BD ⊆ T.
Step 4: E ⊆ T.
Hence, f: ACE is redundant.

Solution (II): (for f:BD E)……
Step 1 : F’={AB ,CD ,ACE} # F’=F-f
Step 2 : T=BD # Set T=determinant
of BDE.
Step 3 : Nothing can be added to set T.
Hence, f: BDE is not redundant .

Closure set of functional dependency :
A closure set of a functional dependency is a set
of
all possible functional dependency that can be derived from a
given set of functional dependencies.
• it is also known as complete set of functional dependencies.
•If F is used to denote the set of FDs for relation R then closure
set of FDs implied by F, it is denoted by F+ .

Algorithm to find closure set of FD:
• Given a set of attributes A, define the closure of A for F as
the set of attributes that are functionally determined by A
under F (find A+ ) .
result = A
while (changes to result) do..
for each XY in F do…
begin
if ….X ⊆ result….then,
result=result ⋃ Y
end

Example to calculate the closure set of FD:
Example 1.Given the following result R={A,B,C}.A set of FD is as
follow:
AB
BC
find out closure of A,B and C? (find A+ ,B+ and C+ )
Solution : (i) A+ = A # A+ = A
= AB # A ⊆ A+ so B ⋃ A+
= ABC # B ⊆ A+ so C ⋃ A+
(ii) B+ = B # B+ =B
= BC #B ⊆ B+ so C ⋃ B+
(iii) C+ = C #C+ =C

Example 2 : Given the following result R={A,B,C,D}.A set of FD is
as follow:
AB, BC,ABD
find closure for A and AB.
Solution : A+ = A #A+ = A
= AB #A ⊆ A+ so B ⋃ A+
= ABC #B ⊆ A+ so C ⋃ A+
= ABCD #AB ⊆ A+ so D ⋃ A+
(AB)+ =AB # (AB)+ = AB
=ABD # AB ⊆ (AB)+ so D ⋃ (AB)+
=ABCD # B ⊆ (AB)+ so C ⋃ (AB)+

Decomposition :
The process of breaking down the given relatio
into two or more relations is known as decomposition.
• Here relation R is replaced by another relation in such a way that…
1. Each new relation contains a subset of the attributes of R
and
2. Together, they all include all tuples and attributes of R.
• example:
R: Account(a_no,balance,B_name,B_city)
R1: Account(a_no,balance,b_name)
R2: Branch(b_name,b_city)

•There are two types of decomposition:
1. lossy decomposition
2. lossless decomposition
1.lossy decomposition:
The decomposition of R into R1 and R2
is lossy when the joint of R1 and R2 does not give the same
relation as in R.
2.Lossless decomposition:
The decomposition of R into R1 and R2
is lossless when the joint of R1 and R2 produces the same
relation as in R.

Example of lossy decomposition:
Balance B_name
5000 Vvn
2000 kkn
8000 kkn
Main table:
decomposition
T1
T2
A_no Balance B_name
A01 5000 Vvn
A02 2000 Kkn
A02 8000 kkn
A03 2000 Kkn
A03 8000 kkn
Joined table:
The resultant joined table
from T1 and T2
which is not same as the
main table.
A_no balance B_name
A01 5000 Vvn
A02 2000 kkn
A03 8000 Kkn
A_no Balance
A01 Vvn
A02 Kkn
A03 kkn

A_no balance B_name
A01 5000 Vvn
A02 2000 kkn
A03 8000 Kkn
A_no B_name
A01 Vvn
A02 kkn
A03 kkn
A_no Balance
A01 5000
A02 2000
A03 8000
A_no balance B_name
A01 5000 Vvn
A02 2000 kkn
A03 8000 kkn
Example of lossless decomposition:
Main table:
T1
T2
Joined table:
The resultant joined table
from T1 and T2
which is same as the
main table.
decomposition

Functional dependencies in Database Management System

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Functional dependencies in Database Management System