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Trivial and Non-Trivial Functional Dependency

 

But the reverse is not true, i.e. if (X ⊇ Y) then X → Y true but if X → Y is true then (X ⊇ Y)  or X ⊉ Y any case occurred

 

Definition of Trivial FD:  If a functional dependency (FD) X → Y holds, where is a subset of X, then it is called a trivial FD.

 

Example:

1. Sid ⊇ Sid so, Sid → Sid

2. SidSname ⊇ Sid so, SidSname → Sid

3. SidSname ⊇ Sname so, SidSname → Sname

4. CidSname ⊇ Snameso, CidSname → Sname

 

Example

AB → A      

   

 R                                                                      

A

B

A

1

1

1

2

1

2

1

3

1

 

In that case, AB → A  is possible because case 3 never possible.

 

Non-Trivial Functional Dependency:
We can define Non-trivial functional dependency in two ways

 

1.  X ⋂ Y = ɸ 

      OR

2.  X ⊉ Y 

 

Example for condition 1:

Sid → Cname - There is no common attribute between Sid and Cname because both are single unique attribute so, it satisfies condition 1  Sid ⋂ Cname = ɸ

 

So, if  Sid → Cname is Functional Dependency and  Sid ⋂ Cname = ɸ then it is non-trivial F.D.

 

See another case:

SidSname → Sname but in this case SidSname ⋂ Sname ≠ ɸ

So, it is Trivial FD but not non-trivial FD.

 

X ⊉ Y: Condition 2

Condition 2, X ⊅ Y it means if X → Y is FD and X ⊇ Y then it is trivial and if X → Y is FD but X ⊉ Y then it is non-trivial FD

Basically, X → Y is FD, and X ⊇ Y id trivial and except trivial all other FDs are non-trivial.

 

Non-trivial: If an FD X → Y holds, where Y is not a subset of X, then it is called a non-trivial FD.

 

Completely non-trivial: If an FD X → Y holds, where X ⋂ Y = ɸ, it is said to be a completely non-trivial FD.

 

Q. Identify all non-trivial FDs

 

A

B

C

1

4

2

2

5

2

3

4

2

3

5

2