Trivial and Non Trivial Functional Dependency examples

Trivial and Non Trivial Functional Dependency examples –

Trivial Functional Dependency – 

Some functional dependencies are said to be trivial because they are satisfied by all relations. Functional dependency of the form A–>B is trivial if  . or A trivial Functional Dependency is the one where RHS is a subset of LHS.

Example, A-->A is satisfied  by all relations involving attribute A.
SSN-->SSN
PNUMBER-->PNUMBER
SSN PNUMBER -->PNUMBER
SSN PNUMBER --> SSN PNUMBER

Non Trivial Functional Dependency – 

Non Trivial Functional Dependency can be categorized into –

• Complete Non Trivial Functional Dependency
• Semi Non Trivial Functional Dependency

Complete Non Trivial Functional Dependency – A Functional Dependency is completely non trivial if none of the RHS attributes are part of the LHS attributes.

Example, SSN --> Ename,
PNUMBER --> PNAME
PNUMBER--> BDATE    X

Semi Non Trivial Functional Dependencies – A Functional Dependency is semi non trivial if atleast one of the RHS attributes are not part of the LHS attributes. {TRIVIAL + NONTRIVIAL}

Question 1 :

A B C 1 1 1 1 2 1 2 1 2 2 2 3

Identify Non Trivial Functional Dependency ?

Solution :

S.NO Dependencies Non Trivial FD ? 1 A→B  × 2 A→C  × 3 A→BC  × 4 B→A  × 5 B→C  × 6 B→AC  × 7 C→A  √ 8 C→B  × 9 C→AB  × 10 AB→C  √ 11 BC→A  √ 12 AC→B  ×

A→B is not a non trivial FD because, for 2, it has two outputs. i.e 2→2 and 2→3.

for AB→C, 11→1, 12→1, 21→2, 22→3, so Non trivial.

Question 2:
 R(A B C D)
 AB {Candidate Key}
 A→C
 B→D .
 Where is the redundancy exist ?

Solution: (A C) and (B D) is suffering from redundancy.

Question 3:
Consider a relation with schema R(A,B,C,D) and FDs {AB -> C, C -> D, D -> A}.
a. What are some of the nontrivial FDs that can be inferred from the given FDs?

Some examples:

C -> ACD
D -> AD
AB -> ABCD
AC -> ACD
BC -> ABCD
BD -> ABCD
CD -> ACD
ABC -> ABCD
ABD -> ABCD
BCD -> ABCD

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