Prime and Non Prime Attributes in DBMS with Example

Prime and Non Prime Attributes in DBMS with Example –

  • Prime Attributes – Attribute set that belongs to any candidate key are called Prime Attributes.
    (union of all the candidate key attribute)
    {CK1 ∪ CK2 ∪ CK3 ∪ ……}
    If Prime attribute determined by other attribute set, then more than one candidate key is possible. For example,
    If A is Candidate Key, and X→A, then, X is also Candidate Key .
  • Non Prime Attribute – Attribute set does not belongs to any candidate key are called Non Prime Attributes.

Some Questions Based on Prime Attributes and Non Prime Attributes –

Question 1 : 
Given a relation R(ABCDEF) having FDs 
  {AB→C, C→D, D→E, F→B, E→F}
Identify the prime attributes and non prime attributes.
Solution :
 (AB)+ : {ABCDEF} ⇒ Super Key
 (A)+  : {A}      ⇒ Not Super Key
 (B)+  : {B}      ⇒ Not Super Key
 Prime Attributes  : {A,B}
 (AB) → Candidate Key
   ↓    (as F → B)
 (AF)+ : {AFBCDE} 
 (A)+  : {A}      ⇒ Not Super key
 (F)+  :  {FB}    ⇒ Not Super Key
 (AF) → Candidate Key
   ↓
 (AE)+ : {AEFBCD}
 (A)+  : {A}      ⇒ Not Super key
 (E)+  : {EFB}    ⇒ Not Super key
 (AE) → Candidate Key
   ↓
 (AD)+ : {ADEFBC}
 (A)+  : {A}      ⇒ Not Super key
 (D)+  : {DEFB}   ⇒ Not Super key
 (AD) → Candidate Key
   ↓
 (AC)+ : {ACDEFB}
 (A)+  : {A}      ⇒ Not Super Key
 (C)+  : {DCEFB}  ⇒ Not Super Key
 ⇒ Candidate Keys {AB, AF, AE, AD, AC}
 ⇒ Prime Attributes {A,B,C,D,E,F}
 ⇒ Non Prime Attributes {} 
Question 2:  
Given a relation R(ABCDEF) having FDs
 {AB → C, C → DE, E → F, C → B}
Identify the prime attributes and non prime attributes.

Solution :
 (AB)+  : {A B C D E F}
 (A)+   : {A}
 (B)+   : {B}
 (AB)  ⇒ (AC), (AC)+ : {ABCDEF}
 (C)+   : {DECBF}
 ⇒ Candidate Keys {AB,  AC}
 ⇒ Prime Attributes {A,B,C}
 ⇒ Non Prime Attributes {D,E,F}

Question 3: 
Given a relation R(ABCDEFGHIJ) having FDs
 {AB → C, A → DE, B → F, F → GH, D → IJ}
Identify the prime attributes and non prime attributes.

Solution : 
 (AB)+ : {ABCDEFGHIJ}
 (A)+  : {DEIJA}
 (B)+  : {FGHB}
 ⇒ Candidate Keys {AB}
 ⇒ Prime Attributes {A,B}
 ⇒ Non Prime Attributes {C,D,E,F,G,H,I,J}
Question 4: 
Given a relation R(ABDLPT) having FDs
 {B → PT, A → D, T → L}
Identify the prime attributes and non prime attributes.

Solution :
 (AB)+ : {ABPTDL}
 (A)+  : {DA}
 (B)+  : {BPTL}
 ⇒ Candidate Keys {AB}
 ⇒ Prime Attributes {A,B}
 ⇒ Non Prime Attributes {D,L,P,T}
Question 5:
Given a relation R(ABCDEFGH) having FDs
 {E → G, AB → C, AC → B, AD → E,B → D, BC → A}
Identify the prime attributes and non prime attributes.

Solution :
 (ABFH)+ : {ABCDEFGH}
 (A)+    : {A} 
 (B)+    : {BD}
 (F)+    : {F}
 (H)+    : {H}
 (AB)    : {ABCDEG}
 (AF)    : {AF}
 (AH)    : {AH}
 (BF)    : {BFD}
 (BH)    : {BHD}
 (FH)    : {FH}
 (ABF)   : {ABFCDEG}
 (ABH)   : {ABHCDEG}
 (AFH)   : {AFH}
 (BFH)   : {BDFH}
None of the proper sets of {ABFH} will determine all the attributes. 
So {ABFH} ⇒ minimal super key or candidate key
 
 (A  B  FH)
     ↓
 (A (AC) FH) ⇒ (ACFH)

 ( A  BFH)
   ↓
 ((BC) BFH) ⇒ (BCFH) 
 ⇒ Candidate Keys {ABFH, ACFH, BCFH}
 ⇒ Prime Attributes {A,B,C,F,H}
 ⇒ Non Prime Attributes {D,E,G}
Question 6 :
Given a relation R(ABCDE) having FDs
 {A → BC, CD → E, B → D, E → A}
Identify the prime attributes and non prime attributes.
Solution :
 (A)+  : {ABCDE}  ⇒ (Candidate Key)
 (E)+  : {ABCDE}  ⇒ (Candidate Key)
 ⇒ Candidate Keys {A,E}
 ⇒ Prime Attributes {A,E}
 ⇒ Non Prime Attributes {B,C,D}
Question 7 :
Consider a relation scheme R(ABCDEH) on which the following Functional Dependencies hold:
 {A → B, BC → D, E → C, D → A}.
What are the candidate keys of R and identify the prime attributes and non prime attributes.
Solution : 
 (BEH)+ : {BEHCDA} ⇒ Super Key
 (B)+    : {B}
 (E)+    : {EC}
 (H)+    : {H}
 (BE)+   : {BECDA}
 (BH)+   : {BHD}
 (EH)+   : {EHC}

None of the proper sets of {BEH} will determine all the attributes. 
So, {BEH} ⇒ minimal super key or candidate key.

(B EH)
 ↑
(A EH)
 ↑
(D EH)
 ⇒ Candidate Keys {BEH, AEH, DEH}
 ⇒ Prime Attributes {A,B,D,E,H}
 ⇒ Non Prime Attributes {C}
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