Discussion
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 ?
Answer: D
The necessary and sufficient combination is Augmentation and Transitivity.
A proof of the Union rule from X → Y and X → Z:
- Augment X → Y with X: XX → XY, that is X → XY.
- Augment X → Z with Y: XY → YZ.
- Transitivity on X → XY and XY → YZ gives X → YZ.
So augmentation and transitivity suffice, and reflexivity is not needed. Both are necessary: transitivity alone can only chain dependencies that already exist and can never put Y and Z together on one right-hand side, and without transitivity the two augmented dependencies cannot be joined. Reflexivity and augmentation alone cannot link X → XY to XY → YZ. All three rules work, but the question asks for the combination that is also necessary.
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