Write Short Notes on Armstrong's Axioms

In database theory, Functional Dependency (FD) is written as $X \to Y$, where $X$ and $Y$ are attribute sets. To reason about which FDs logically follow from a given set, we use Armstrong's axioms—a small sound and complete set of inference rules for FDs.
The axioms are:

1. Reflexivity: if $Y \subseteq X$ then $X \to Y$.
2. Augmentation: if $X \to Y$ then for any $Z$, $XZ \to YZ$.
3. Transitivity: if $X \to Y$ and $Y \to W$ then $X \to W$.

These three are the core inference mechanisms; many useful derived rules can be obtained from them.

Note: The “soundness” property means every FD derived using the axioms is logically implied by the original FDs; “completeness” means any logically implied FD can be derived from them.

Write-short-notes version (what to include)

A good short note typically lists (i) definitions, (ii) the axioms, and (iii) at least 1–2 derived rules used in exams.

Key objects you should mention:

• Attribute set: e.g., $X, Y, Z \subseteq U$.
• Closure: helps compute what follows from a set of FDs.
• Implied FD: written $\Sigma \models f$.

Armstrong's axioms are inference rules used to derive implied FDs:

• Reflexivity: $Y \subseteq X \Rightarrow X \to Y$
• Augmentation: $X \to Y \Rightarrow XZ \to YZ$
• Transitivity: $(X \to Y) \wedge (Y \to W) \Rightarrow X \to W$

Common derived rules (often expected in “short note” answers)

Derived rules are consequences of Armstrong’s axioms and are frequently used to simplify derivations:

Union rule
If $X \to Y$ and $X \to Z$, then $X \to YZ$.

Decomposition rule
If $X \to YZ$, then $X \to Y$ and $X \to Z$.

Pseudotransitivity
If $X \to Y$ and $WY \to Z$, then $XW \to Z$.

Quick derivation idea for Union (sketch)

Using augmentation and transitivity, you can combine $X \to Y$ and $X \to Z$ into $X \to YZ$ by first augmenting one FD so that the RHS aligns with the other’s determined attributes, then chaining.