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Explain closure of attributes in DBMS
Closure of an attribute x is the set of all attributes that are functional dependencies on X with respect to F. It is denoted by X+ which means what X can determine.
Algorithm
Let’s see the algorithm to compute X+
- Step 1 − X+ =X
- Step 2 − repeat until X+ does not change
- For each FD Y->Z in F
- If Y ⊆ X+ then X+ = X+ U Z
- For each FD Y->Z in F
Example 1
Consider a relation R(A,B,C,D,E,F)
F: E->A, E->D, A->C, A->D, AE->F, AG->K.
Find the closure of E or E+
Solution
The closure of E or E+ is as follows −
E+ = E
=EA {for E->A add A}
=EAD {for E->D add D}
=EADC {for A->C add C}
=EADC {for A->D D already added}
=EADCF {for AE->F add F}
=EADCF {for AG->K don’t add k AG ⊄ D+)Example 2
Let the relation R(A,B,C,D,E,F)
F: B->C, BC->AD, D->E, CF->B. Find the closure of B.
Solution
The closure for B is as follows −
B+ = {B,C,A,D,E}
Closure is used to find the candidate keys of R and compute F+
Candidate key of R: X is a candidate keys of R if X->{R}
For example,
R(A,B,C,D,E,F) WHERE F:A->BC, B->D, C->DE, BC->F. Then, find the candidate keys of R.
Solution
A+= {A,B,C,D,E,F}={R}=>A is a candidate key
B+= {B,D} => B is not a candidate key
C+= {C,D,E} => C is not a candidate key
BC+= {B,C,D,E,F} => BC is not a candidate key
Closure of F (F+): F+ is the set of all FDs that can be inferred/ derived from F. Using Armstrong Axioms repeatedly on F, we can compute all the FDs.
Example
R(A,B,C,D,E) AND F: A->B,B->C, C->D, A->E. Find the closure of F
Solution
A+= {A,B,C,D,E}
B+= {B,C,D}
C+= {C,D}
F+= {A->A, A->B, A->C, A->D, A->E, B->B, B->C, B->D, C->C, C->D}
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