## What is a Multivalued Dependency ?

To understand the concept of MVD, let us consider a schema denoted as * MPD (Man, Phones, Dog_Like)*,

Person : |
Meaning of the tuples |
|||

Man(M) |
Phones(P) |
Dogs_Like(D) |
⇒ | Man M have phones P, and likes the dogs D. |

M1 | P1/P2 | D1/D2 | ⇒ | M1 have phones P1 and P2, and likes the dogs D1 and D2. |

M2 | P3 | D2 | ⇒ | M2 have phones P3, and likes the dog D2. |

Key : MPD |

There are no non trivial FDs because all attributes are combined forming Candidate Key i.e. MDP. The multivalued dependency is shown by * “→→”*. So, in the above relation, two multivalued dependencies exists –

**Man →→ Phones****Man →→ Dogs_Like**

A man’s phone are independent of the dogs they like. But after converting the above relation in Single Valued Attribute, Each of a man’s phones appears with each of the dogs they like in all combinations.

Man(M) |
Phones(P) |
Dogs_Likes(D) |

M1 | P1 | D1 |

M1 | P2 | D2 |

M2 | P3 | D2 |

M1 | P1 | D2 |

M1 | P2 | D1 |

**Some Points to note here about relation Person :**

- Some unwanted(shaded) tuples will also exist in the relation while converting it into single valued attributes.
- However, We can see that the relation is in BCNF, and thus we would not consider decomposing if further if we looked only at the FDs that hold over the relation Person.
- The
**redundancy exists in BCNF relation because of MVD**.

**Where MVD occurs ?**

- If two or more independent relations are kept in a single relation, then Multivalued Dependency is possible.

For example, Let there are two relations :**Student(SID, Sname)**where**(SID → Sname)****Course(CID, Cname)**where,**(CID → Cname)**

There is no relation defined between Student and Course.

If we kept them in a single relation named, then MVD will exists because of**Student_Course***m:n Cardinality*.**Student :****SID****Sname**S1 A S2 B **Course :****CID****Cname**C1 C C2 B **Merging using Cross Product –**

As Student and Course do not have any relation, So taking

all the possible combinations by using Cross product**SID****Sname****CID****Cname**S1 A C1 C S1 A C2 B S2 B C1 C S2 B C2 B **2 Multivalued Dependency exists :**

1. SID →→ CID

2. SID →→ Cname

- If two or more multivalued attributes exists in a relation, then while converting into single valued attributes, MVD exists. The relation
is such type of example.**“Person”**

##### Definition of MVD :

Let * R* be the relational schema,

*be the attribute sets over*

**X,Y***. A MVD*

**R***exists on a relation*

**(X→→Y)***:*

**R**If two tuples

*and*

**t**_{1}*exists in*

**t**_{2}*, such that*

**R***then two tuples*

**t**_{1}[X] = t_{2}[Y]*and*

**t**_{3}*should also exist in*

**t**_{4}*with the following properties where*

**R**

**Z = R – {X ∪ Y}:**- t
_{3}[X] = t_{4}[X] = t_{1}[X] = t_{2}[X] - t
_{3}[Y] = t_{1}[Y] and t_{4}[Y] = t_{2}[Y] - t
_{3}[Z] = t_{2}[Z] and t_{4}[Z] = t_{1}[Z]

The tuples * t_{1}, t_{2}, t_{3}, t_{4}* are not necessarily distinct.

#### Inference Rules of MVD (Five Rules)

Three of the additional rules involve only MVDs :

C- |
Complementation |
: | If X →→ Y, then X →→ {R − (X∪Y)} . |

A- |
Augmentation |
: | If X →→ Y and W ⊇ Z, then WX →→ YZ. |

T- |
Transitivity |
: | If X →→ Y and Y →→ Z, then X →→ (Z − Y ). |

The remaining two rules relate FDs and MVDs :

**Replication :**If X → Y, then X →→ Y but the reverse is not true.**Coalescence :**If X →→ Y and there is a W such that W ∩ Y is empty, W → Z, and Y ⊇ Z, then X → Z.

#### Trivial and Non Trivial MVD :

A MVD * X →→ Y* in

*is called a trivial MVD is*

**R**is a subset of**Y**or**X (X ⊇ Y)**. Otherwise, it is a non trivial MVD and we have to repeat values redundantly in the tuples.**X ∪ Y = R**

#### Removal of MVD :

**Solution :** Fourth Normal Form (4NF) – Click for removal of MVD.

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BCNF – Boyce Codd Normal Form | 4NF Normal Form |

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