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Functional Dependency

Minakshi.

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To determine F+, we need rules for deriving all functionaldependencies that are implied by F. A set of rules that may beused to infer additional dependencies was proposed by

Armstrong in 1974. These rules (or axioms) are a complete setof rules in that all possible functional dependencies may bederived from them.

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For example, the studentrelation has the

following functional dependencies

sno -> sname cno -> cname sno -> address cno ->instructor instructor -> office

Let these dependencies be denoted byF. The

closure ofF, denoted byF+, includes Fand all

functional dependencies that are implied byF.

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These rules are called Armstrong'sAxioms.

Reflexivity Rule --- If X is a set of attributes and Y is a subsetof X, then X -> Yholds. The reflexivity rule is the mostsimple (almost trivial) rule. It states that each subset of X isfunctionally dependent on X.

Augmentation Rule --- IfX -> Yholds and W is a set of

attributes, thenWX -> WYholds. The augmentation rule isalso quite simple. It states that if Y is determined by X then aset of attributes W and Y together will be determined by Wand X together. Note that we use the notation WX to meanthe collection of all attributes in W and X and write WX ratherthan the more conventional (W, X) for convenience.

Transitivity Rule --- IfX -> YandY -> Z hold, then X -> Zholds. The transitivity rule is perhaps the most important one.It states that if X functionally determines Y and Y functionallydetermines Z then X functionally determines Z.

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Further axioms may be derived from the above although the abovethree axioms are sound and complete in that they do not generateany incorrect functional dependencies (soundness) and they dogenerate all possible functional dependencies that can be inferredfrom F(completeness). For proof of soundness and completeness ofArmstrong's Axioms, the reader is referred to Ullman (Vol 1, page387). The most important additional axioms are:

Union Rule --- IfX -> YandX -> Zhold, thenX -> YZholds.

Decomposition Rule --- IfX -> YZholds, then so doX -> YandX ->Z.

Pseudotransitivity Rule --- IfX -> Yand WY -> Zhold then so doesWX -> Z.

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Examples

sno, cno) -> sno (Rule 1)

(sno, cno) -> cno (Rule 1)

(sno,c

no)->

(sname,c

name) (Rule 2)cno -> office (Rule 3)

sno -> (sname,address) (Union Rule)

etc.

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Closure of a set of Functional

dependenciesConsider a relation

R=(A,B,C,G,H,I)

anda set of Functional dependencies

A B

A CCG H

CG I

B H

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some members ofF+

Ap H by transitivity fromAp B andBp H

AGp I by augmentingAp Cwith G, to getAGp CG

and then transitivity with CGp I

CGp HI

by augmenting CGp Ito inferCGp CGI,and augmenting ofCGp Hto inferCGIp HI,

and then transitivity

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

Compute the closure of the following set of

functional dependencies A BC

CD E

B D E A

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R = (A,B,C, G,H,I)

F = {Ap B

Ap C

CGp H

CGp I

Bp H}

Compute (AG)+

Example of Attribute Set Closure

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