📘 Armstrong’s Axioms (Inference Rules)

Armstrong’s axioms are a set of rules used to infer new functional dependencies (FDs) from a given set of dependencies F.

👉 They help us compute closure (F⁺) — all possible dependencies that logically follow from F.

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🌟 Key Properties

✅ Soundness

• Any FD derived using these rules is always correct
• It will hold in every valid relation that satisfies F

✅ Completeness

• Using these rules, we can derive all possible FDs

• That means:

  Armstrong’s axioms are sufficient to compute F⁺ (closure of F)

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⚙️ Basic Armstrong’s Axioms (Core Rules)

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🔹 IR1: Reflexivity Rule

📌 Statement:

If:

X ⊇ Y

Then:

X → Y

💡 Meaning:

• A set of attributes determines itself and its subsets

🔸 Type:

• Produces trivial functional dependencies

✔ Example:

{A, B} → A

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🔹 IR2: Augmentation Rule

📌 Statement:

If:

X → Y

Then:

XZ → YZ

💡 Meaning:

• If a dependency holds, adding the same attributes to both sides keeps it valid

✔ Example:

A → B  
⇒ AC → BC

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🔹 IR3: Transitivity Rule

📌 Statement:

If:

X → Y  
Y → Z

Then:

X → Z

💡 Meaning:

• Functional dependencies can be chained

✔ Example:

Dept_no → Mgr_ssn  
Mgr_ssn → Mgr_phone  
⇒ Dept_no → Mgr_phone

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📌 Important Concepts

🔸 Trivial vs Non-trivial FD

• Trivial FD:

  X → Y where Y ⊆ X
  

• Non-trivial FD:

  Y ⊄ X
  

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🔁 Derived Rules (From Armstrong’s Axioms)

These rules are not basic, but can be derived using IR1–IR3.

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🔹 IR4: Decomposition Rule

📌 Statement:

X → YZ ⇒ X → Y and X → Z

💡 Meaning:

• Break RHS into smaller parts

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🔹 IR5: Union Rule

📌 Statement:

X → Y and X → Z ⇒ X → YZ

💡 Meaning:

• Combine multiple dependencies into one

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🔹 IR6: Pseudotransitive Rule

📌 Statement:

X → Y and WY → Z ⇒ WX → Z

💡 Meaning:

• A generalized form of transitivity with extra attributes

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⚠️ Important Notes

• These rules must be applied carefully
• Common misunderstandings:

❌ Not Always True:

X → A and Y → B  ≠  XY → AB

❌ Not Always True:

XY → A  ≠  X → A or Y → A

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How Designers Use Armstrong’s Axioms

1. Start with given FDs (F)
2. Apply IR1, IR2, IR3 repeatedly
3. Derive:
   • New FDs
   • Attribute closures
4. Use results for:
   • Normalization
   • Schema design
   • Eliminating redundancy

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 Summary

“Armstrong’s axioms are a complete and sound set of rules used to infer all possible functional dependencies from a given set, forming the foundation for database normalization and design.”

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🏁 Quick Recap Table

Rule	Name	Idea
IR1	Reflexivity	Subset dependency
IR2	Augmentation	Add attributes to both sides
IR3	Transitivity	Chain dependencies
IR4	Decomposition	Split RHS
IR5	Union	Combine RHS
IR6	Pseudotransitive	Extended transitivity