6.13- Armstrong's Axioms in dbms in HINDI | Armstrong rules database | Closure of Attributes

6.13- Armstrong's Axioms in dbms in HINDI | Armstrong rules database | Closure of Attributes

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Relational Database Design- Functional Dependencies And Normalization


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6.5.1 Armstrong’s Axioms
The following three rules called inference axioms or Armstrong’s Axioms can be used to find all the FDs logically implied by a set of FDs. Let X, Y, Z, and W be subsets of attributes
of a relation R. The following axioms hold:
1. Reflexivity. If Y is a subset of X, then X → Y. This also implies that X → X always
holds. Functional dependencies of this type are called trivial functional dependencies.
2. Augmentation. If X → Y holds and Z is a set of attributes, then ZX → ZY.
3. Transitivity. If X → Y holds and Y → Z holds, then X → Z holds.
These rules are Sound and Complete. They are sound because they do not generate any
invalid functional dependencies. They are complete because they allow us to generate F+
Functional Dependency and Normalisation 201
(closure of F) from the given set of functional dependencies F. It is very cumbersome and
complex to use Armstrong’s Axioms directly for the computation of F+
. So some more axioms are added to simplify the process of computation of F+
. These additional axioms can
be proved correct by using Armstrong’s Axioms. The additional axioms are
4. Additivity or Union. If X → Y and X → Z, then X → YZ holds.
5. Projectivity or Decomposition. If X → YZ holds, then X → Y and X → Z also holds
6. Pseudotransitivity. If X → Y and ZY → W holds, then XZ → W holds.
These additional axioms are also Sound and Complete.
Example. Let R = (A, B, C, D) and F be the set of functional dependencies for
R given by {A→B, A→C, BC→D}. Prove A→D.
Sol. Given set of functional dependencies for a relation R is {A→B, A→C, BC→D}
By using Armstrong Axioms, named projectivity, we can show that
A→BC (as A→B, A→C)
Since BC→D, so by transitivity rule,
A→BC and BC→D means A→D. Hence proved.
Example. Consider a relation R2(A, B, C, D, E, F) and a set of functional dependencies
FD={AB→C, C→FA, F→E} that hold on R2.
(i) Using Armstrong’s axioms show that the functional dependency AB→E also holds
on R.
(ii) Does AB→F hold for FD1 = {AB→CD, C→E, DE→F}?
Sol.
(i) AB→C (given).
Apply decomposition to C→FA, get C→F.
F→E (given).
Apply transitivity to AB→C, C→F, F→E, get AB→E. Hence proved.
(ii) Yes. We can prove it as follows:
AB→CD (given).
C→E (given). Apply augmentation to C→E, get, CD→DE.
DE→F (given).
Apply transitivity to AB→CD, CD→DE and DE→F, get AB→F.
Example. Let F = {AB → C, B → D, CD→E, CE→GH}. Give a derivation sequence on
FD, {AB→G} using only Armstrong’s axioms.
Sol. We have to derive AB → G
AB → ABE (augmentation: AB → E with AB)
ABE → CE (augmentation: AB → C with E)
AB → CE (transitivity: AB → ABE and ABE → CE)
AB → GH (transitivity: AB → CE and CE → GH)
GH → G (reflexivity)
AB → G (transitivity: AB → GH and GH → G)

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