Database Normalization Solved Exercise 7

Set of solved exercises in Normalization / Normalization Solved Examples / How to find candidate keys, and primary keys in database? / Sets of examples to find the keys of a tables / Process of Key finding in a database – Examples

Question:

Consider the relation scheme R = (E, F, G, H, I, J, K, L, M, N) and the set of functional dependencies {{E, F} → {G}, {F} → {I, J}, {E, H} → {K, L} → {M}, {K} → {M}, {L} → {N}} on R. What is the key for R?

Solution:

To find the key of a relation, we need to find the closure of attributes. If any attribute’s or set of attributes’ closure gives all the attributes of the relation, then we would say that attribute/set of attributes as the key for that relation.

To simplify this task or to avoid finding closure of all attributes, let us do find the closure for left hand side (LHS) attributes of the functional dependencies.

For the given question, attributes F, (E, F), (E, H), K, and L are the LHS attributes.

• The closure of F, i.e., (F)⁺ = FIJ ≠ EFGHIJKLMN. Hence F is not a candidate key. [Refer, How to find closure]

• (EF)⁺ = EFGIJ ≠ EFGHIJKLMN. Hence EF is not a candidate key.

• (EH)⁺ = EHKLMN ≠ EFGHIJKLMN. Hence EH is not a candidate key.

• (K)⁺ = KM ≠ EFGHIJKLMN. Hence, K is not a candidate key.

We observed that the left side attributes of any FDs alone cannot form a key. Let us try with the combination of two FDs LHS.

• (EFH)⁺ = EFGHIJKLMN  = R.

Hence, EFH is the key for this relation.

[

Derivation of EFH+:

Step 1: result = EFH

Step 2: result = EFGH from {E, F} → {G}

Step 3: result = EFGHIJ from {F} → {I, J}

Step 4: result = EFGHIJKLM from {E, H} → {K, L} → {M}

Step 5: result = EFGHIJKLMN from {L} → {N}

We have reached that the result = R. hence, EFH is the key.

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Related Posts:

Various uses of attribute closure

How to find Closure?

How to find candidate keys using closure finding algorithms