6.14 How To Find The Closure Set Of Functional Dependencies With Example | Normal Forms | dbms

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Relational Database Design Functional Dependencies And Normalization

How To Find The Closure Set Of Functional Dependencies With Example,
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Closure of a Set of Functional Dependencies
Assume that F is a set of functional dependencies for a relation R. The closure of F, denoted
by F+
, is the set of all functional dependencies obtained logically implied by F i.e., F+
is
the set of FD’s that can be derived from F. Furthermore, the F+ is the smallest set of FD’s
such that F+ is superset of F and no FD can be derived from F by using the axioms that
are not contained in F+
.
If we have identified all the functional dependencies in a relationship then we can easily
identify superkeys, candidate keys, and other determinants necessary for normalization.
Algorithm: To compute F+
, the closure of FD’s
Input: Given a relation with a set of FD’s F.
Output: The closure of a set of FD’s F.
Step 1. Initialize F+
= F // F is the set of given FD’s
Step 2. While (changes to F+
) do
Step 3. For each functional dependency f in F+
Apply Reflexivity and augmentation axioms on f and add the resulting functional
dependencies to F+
.
Step 4. For each pair of functional dependencies f1 and f 2 in F+
Apply transitivity axiom on f1 and f2
If f 1 and f 2 can be combined add the resulting FD to F+
.
Step 5. For each functional dependency to F+
Apply Union and Decomposition axioms on f and add the resulting functional
dependencies to F+
.
Step 6. For each pair of functional dependencies f1 and f 2 in F+
Apply Pseudotransitivity axiom on f1 and f 2
If f1 and f2 can be combined add the resulting FDs to F+
.
The additional axioms make the process of computing F+
easier by simplifying the
the process used for steps 3 and 4. If we want to compute F+
only by using Armstrong’s rule
than eliminate steps 5 and 6.
Example. Consider the relation schema R = {H, D, X, Y, Z} and the functional dependencies
X→YZ, DX→W, Y→H
Find the closure F+ of FD’s.
Sol. Applying Decomposition on X→YZ gives X→Y and X→Z
Applying Transitivity on X→Y and Y→H gives X→H
Thus the closure F+
has the FD’s
X→YZ, DX→W, Y→H, X→Y, X→Z, X→H
Example. Consider the relation schema R= {A, B, C, D, E} and Functional dependencies
A→BC, CD→E, B→D, E→A
Sol. Applying Decomposition on A→BC gives A→B and A→C.
Functional Dependency and Normalisation 203
Applying Transitivity on A→B and B→D gives A→D.
Applying Transitivity on CD→E and E→A gives CD→A
Thus the closure F+
has the FD’s
A→BC, CD→E, B→D, E→A, A→B, A→C, A→D, CD→A.

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