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 –

  1. Man →→ Phones
  2. 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 Student_Course, then MVD will exists because of 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 “Person” is such type of example.
Definition of MVD : 

Let R be the relational schema, X,Y be the attribute sets over R. A MVD (X→→Y) exists on a relation R :
If two tuples t1 and t2 exists in R, such that t1[X] = t2[Y] then two tuples t3 and t4 should also exist in R with the following properties where Z = R – {X ∪ Y}:

  • t3[X] = t4[X] = t1[X] = t2[X]
  • t3[Y] = t1[Y] and t4[Y] = t2[Y]
  • t3[Z] = t2[Z] and t4[Z] = t1[Z]

The tuples t1, t2, t3, t4 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 R is called a trivial MVD is

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

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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