In the context of schema normalization in relational DBMS, consider a set F of…
2026
In the context of schema normalization in relational DBMS, consider a set F of functional dependencies. The set of all functional dependencies implied by F is called the closure of F. To compute the closure of F, Armstrong’s Axioms can be applied. Consider 𝑋, 𝑌, and 𝑍 as sets of attributes over a relational schema. The three rules of Armstrong’s Axioms are described as follows
Reflexivity: If 𝑌 ⊆ 𝑋 , then 𝑋 → 𝑌
Augmentation: If 𝑋 → 𝑌, then 𝑋𝑍 → 𝑌𝑍 for any Z
Transitivity: If 𝑋 → 𝑌 and 𝑌 → 𝑍, then 𝑋 → Z
The additional rule of Union is defined as follows.
Union: If 𝑋 → 𝑌 and 𝑋 → 𝑍, then 𝑋 → 𝑌𝑍
It can be proved that the additional rule of Union is also implied by the three rules of Armstrong’s Axioms. Listed below are four combinations of these three rules. Which one of these combinations is both necessary and sufficient for the proof ?
- A.
Reflexivity, Augmentation, and Transitivity
- B.
Reflexivity and Augmentation
- C.
Transitivity
- D.
Augmentation and Transitivity
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Correct answer: D
The correct answer is Option D: Augmentation and Transitivity.
Proof of Union Rule:
Given:
X → Y
X → Z
Step 1:
From X → Y, apply Augmentation with X:
XX → XY
Since XX = X,
X → XY
Step 2:
From X → Z, apply Augmentation with Y:
XY → YZ
Step 3:
Using Transitivity on:
X → XY
XY → YZ
We get:
X → YZ
Thus, the Union rule is derived using only:
Augmentation
Transitivity
Reflexivity is not required in this proof.
Hence, the correct answer is Option D.