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 ?
Correct : d
The Union rule states: if X→Y and X→Z, then X→YZ. To prove this using Armstrong''s Axioms, we need to find the minimal combination of rules that makes this proof possible.
The proof works as follows. Start with X→Z. Apply Augmentation by adding X to both sides — since augmenting X→Z with X gives XX→XZ, and XX simplifies to X, we get X→XZ. Next, take X→Y and apply Augmentation by adding Z to both sides, giving XZ→YZ. Now we have X→XZ and XZ→YZ. Applying Transitivity to these two gives X→YZ, which is exactly the Union rule. The proof is complete using only Augmentation and Transitivity.
Reflexivity is never used in any step of this proof, so option A is incorrect — it includes an unnecessary rule and fails the "necessary" condition. Option B has Reflexivity and Augmentation but without Transitivity there is no way to chain X→XZ and XZ→YZ into X→YZ. Option C has only Transitivity but without Augmentation neither X→XZ nor XZ→YZ can be derived from the given dependencies.
Correct answer: D — Augmentation and Transitivity ✓
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