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**Multivalued dependency** (MVD) has the presence of one or more rows in a table. A multivalued dependency prevents fourth normal form. It involves at least three attributes of a table. It is a constraint between two sets of attributes in a relation.

It is represented with a symbol "->->" in the database management system (DBMS).

**We use multivalued conditions in two different ways −**

To test the relations to decide, if they are lawful under a given arrangement of practical and multivalued dependencies.

To determine limitations on the arrangement of lawful relations. We will concern ourselves just with relations that fulfill a given arrangement of practical and multivalued dependencies.

Consider an example to **demonstrate multivalued dependency** −

Let us consider a relational schema R(A,B,C,D) does the multivalued dependency A →→ BC logically imply the multivalued dependencies A →→ B and A →→ C? If yes, prove it, otherwise give a counterexample.

A --> BC is a notation for A --> B | C

That is, A --> B and also A --> C can hold. (Union rule among inference rules of functional dependency)

The notation A --> BC is always useful because multi-value-dependency always comes in pairs. This implies that the relation equals AB JOIN AC. Ie the relation satisfies join dependency {AB, AC}.

The union rule of inference rule is as follows −

A --> B , A --> C ==> A --> BC

A → B (given)

A → C (given)

A → AB (by augmentation with A. Where AA = A)

AB → BC (by augmentation with B)

A → BC (using transitive rule on 3 and 4)

Let us try to understand the proof of transitive rule for MVD −

If A ->B holds and B ->C holds, then A ->B −>C holds.

The given functional dependency (FD) set is as follows −

ISBN--> TITLE, PUBLISHER

ISBN, NO -->AUTHOR

PUBLISHER -->PU_URL

We need to prove the rule. Consider A=ISBN, B=PUBLISHER, C=PU_URL.

To find the Transitive rule is implied, find the cover of A+ and compute.

Now start with x={ISBN}

The FD ISBN--> TITLE, PUBLISHER has LHS which is completely contained in current attribute set x.

Extend x by FD RHS attribute set, giving x={ISBN,TITLE,PUBLISHER}

Now FD:PUBLISHER -->PU_URL is applicable

Add RHS attribute set of FD to current attribute SET x, giving x={ISBN,TITLE,PUBLISHER,PU_URL}

Here we can conclude that ISBN-->PU_URL

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