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# Closure Method In DBMS

• Closure method is used to find the all possible candidate keys in the table.
• A candidate key is the minimal-key which can determine all the attributes in the relation.

#### Notation Of Closure Set

• Closure of attribute {X} is denoted as {X}+.
• Closure of {X}+ means what the X may or may not determine all the attributes in the given relation.

Keep in mind: If Closure of any attribute ({X}+) can determines the all attributes of given relation then closure of that attribute can be used as candidate key otherwise not.

## Method to Find Closure of an Attribute Set

• Find all determined values from closure and add recursive value of closure.
• Find the closure and candidate key

Whenever we need to find the closure of any relation the following two things will be given

1. Relation with attributes i.e. {R(ABCDEF)}
2. Functional dependencies (FD) i.e. {A→B, BC→D, E→F} etc.

We need to find the closure of all those attributes which are left side of the FD.

Let explain with multiple Examples

### Example 01

Suppose a relation (R) containing four attributes as A, B, C, and D and functional dependencies as given below.

• R= {A, B, C, D}
• FD = {A→B, B→C, C→D}

As attributes A, B, C and D are present in the left side of FD so, we will find the closure of all these attributes.

So, Closure of all attributes are given under

Closure of attribute “A”

• According to recursive rule, Attribute “A” can determine Attribute “A” itself.
• According to given FD, Attribute “A” can directly determine Attribute “B”.
• According to transitive property, Attribute “A” can determine “C” through “B”.
• According to transitive property, As Attribute “A” already determine C So, Attribute “A” can determine “D” through “C”.

So, Closure of A = A+ = ABCD

Closure of attribute “B”

• According to recursive rule, Attribute “B” can determine Attribute “B” itself.
• According to given FD, Attribute “B” can directly determine Attribute “C”.
• According to transitive property, Attribute “B” can determine “D” through “C”.
• Attribute “B” cannot determine attribute “A”

So, Closure of B = B+ = BCD

Closure of attribute “C”

• According to recursive rule, Attribute “C” can determine Attribute “C” itself.
• According to given FD, Attribute “C” can directly determine Attribute “D”.
• Attribute “C” cannot determine attribute “A” and “B”

So, Closure of C = C+ = CD

Closure of attribute “D”

• According to recursive rule, Attribute “D” can determine Attribute “C” itself.
• Attribute “D” cannot determine attribute “A”, “B” and “C”

So, Closure of D = D+ = D

Conclusion: As we see only the closure of attribute “A” can determine the all attributes of relation so attribute “A” can be used as Candidate key.

So, Candidate Key = {A}

#### Keep In Mind:

Attribute set AB, AC, AD, or ABC, ACD or ABCD can be used to determine all the attributes in the relation but cannot consider as candidate key.

Because  candidate key is a minimal key to determine all attributes in the relation. So “A” is a candidate key and combination of A with others like (AB, AC, AD, or ABC, ACD or ABCD) is considered as Super Key.

## Example 02

Let suppose R= {A, B, C, D} and  FD = {A→B, B→C, C→D, D→A}

As attributes A, B, C and D are present in the left side of FD so, we will find the closure of all these attributes.

Closure of attribute “A”

• As, Attribute “A” can determine itself.
• According to FD, Attribute “A” can directly determine Attribute “B”.
• According to transitive property, Attribute “A” can determine “C” through “B”.
• According to transitive property, As Attribute “A” already determine C So, Attribute “A” can determine “D” through “C”.

So, Closure of A = A+ = ABCD

Closure of attribute “B”

• As, Attribute “B” can determine itself.
• According to FD, Attribute “B” can directly determine Attribute “C”.
• According to transitive property, Attribute “B” can determine “D” through “C”.
• According to transitive property, As Attribute “B” already determine D So, Attribute “B” can determine “A” through “D”.

So, Closure of B = B+ = BCDA

Closure of attribute “C”

• As, Attribute “C” can determine itself.
• According to FD, Attribute “C” can directly determine Attribute “D”.
• According to transitive property, As Attribute “C” already determine D So, Attribute “C” can determine “A” through “D”.
• As Attribute “C” already determine A So, Attribute “C” can determine “B” through “A”.

So, Closure of C = C+ = CDAB

Closure of attribute “D”

• As, Attribute “D” can determine itself.
• According to FD, Attribute “D” can directly determine Attribute “A”.
• According to transitive property, As Attribute “D” already determine A So, Attribute “D” can determine “B” through “A”.
• As Attribute “D” already determine B So, Attribute “D” can determine “C” through “B”.

So, Closure of D = D+ = DABC

Conclusion: As we see the closure of all attributes “A”, “B”, “C” and “D” can determine the all attributes of relation so all attributes can be used as Candidate key.

So, Candidate Key = {A, B, C, D}

### Example 03:  Find the closure and candidate key

Consider a relation R ( A , B , C , D , E , F , G ) with the functional dependencies-

• A → BC
• BC → DE
• D → F
• CF → G

Here we will find the closure of attribute “A”, “BC”, “D”, and “CF” but keep in mind,  First, we have to find the closure of single attribute i.e. (A and D) and then find the closure of  multiple attributes (“BC” and “CF”)

Now, let us find the closure of some attributes and attribute sets-

Closure of attribute A

A+   = {A} According to recursive property, A can determine itself.

Using FD, A → BC,  According to decomposition rule “A” can determine “B” and “C” )

A+   = { A , B , C }

As B, C is already determined in A+ So by, Using BC → DE attributes D and E and also be determined.

A+  = { A , B , C , D , E }

As attribute “D” is determined in closure of “A” so by using  D → F. attribute “F” can also be determined

A+  = { A , B , C , D , E , F }

( Using D → F ) As attributes { A , B , C , D , E , F } is determined in closure of “A” so by using  (CF → G). attribute “G” can also be determined

A+  = { A , B , C , D , E , F , G }    ( Using CF → G )

Thus,

A+ = { A , B , C , D , E , F , G }

Closure of attribute D

D+   = { D }

D+= { D , F }   ( Using D → F )

We cannot determine any other attribute using attributes D and F contained in the result set.

Thus,

D+ = { D , F }

“D” is not a candidate key because it cannot determine the attribute”A”, “B” and “C”.

Closure of attribute set {B, C}

{ BC }+= { B , C }

{ BC }+= { B , C , D , E }               ( Using BC → DE )

{ BC }+= { B , C , D , E , F }          ( Using D → F )

{ BC }+= { B , C , D , E , F , G }    ( Using CF → G )

Thus,

{ BC }+ = { B , C , D , E , F , G }

“BC” is not a candidate key because it cannot determine the attribute ”A”

Closure of attribute set {C,F}

{ CF }+= { C , F }

{ CF }+= { C , F , G  }               ( Using CF → G )

Thus,

{CF }+ = {C, F , G }

“BC” is not a candidate key because it cannot determine the attribute ”A, Candidate key

So the candidate key is only “A” because closure of A determines the all attributes of relations

## Short-cut to find the candidate key

#### Example 4:

Smart trick to find the all possible candidate keys in the relation.  Suppose the following relation

R(A,B,C,D,E)

FD= {A→B, BC→D, E→C, D→A}

1.    If any attributes does not exist right side of FD then it must be the part of candidate key with some other attribute as well.

Check in above given relation that, every attribute is determined from some attributes but attribute E is not determined by any attribute. So attribute “E” must be used in candidate Key with some other Key to determine E itself.

2.  As Closure of EA+ can determines the all attributes of Relation so EA is first candidate key, to make rest of the candidate keys follow the following

I.  Check either E or A attributes, present in the Right hand side of any FD. If E or A founds in Right side of FD then replace with the Left side of that FD.

For second candidate key A is replaced with D and E does not exist to right side of any FD so it remains the same. Therefore the second candidate key will be ED.

II. In the same way now D will check in all FD’s. If D is found in Right hand side of any FD then it can also replace. Keep in mind the concept of decomposition, if there are two attributes are determining the single attribute.

As in this case D will be replacing either with B or C. here we check the closer of EB and EC and select one of them. So EB is the valid candidate key but EC cannot determines the entire attributes so

Candidate keys will be  (EA, ED, BE

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