Chapter 18, Problem 34E

Explanation of Solution

Three Armstrong’s axiom:

The below are three Armstrong’s axiom and they are

• Reflexivity
  • A is considered to be set of attributes
  • $\text{B}\subseteq \text{C and B}\subseteq \text{A then A}\to \text{B}$
• Augmentation
  • A is considered to be set of attributes
  • $\text{B}\to \text{C, then AB}\to \text{AC}$ will also hold
• Transitivity
  • $\text{ A}\to \text{B and B}\to \text{A}$, then $\text{A}\to \text{C}$ can also be hold.

Other axioms:

• Decomposition
  • When $\text{ A}\to \text{BC}$ holds, then $\text{A}\to \text{B}$ and $\text{A}\to \text{C}$ can also hold.
• Union
  • When $\text{ A}\to \text{B}$ and $\text{A}\to \text{C}$ holds, then $\text{A}\to \text{BC}$ can also hold.

Computing the closure of the given functional dependency:

The below is the computation of the functional dependencies (F⁺):

• Considering the dependency $\text{ A}\to \text{BC}$
• Utilizing the decomposition axiom to yield $\text{ A}\to \text{B}$ and $\text{ A}\to \text{C}$
• The transitive axiom is used on the provided functional dependency $\text{ B}\to \text{D}$ and also with the functional dependency $\text{ A}\to \text{B}$ to get $\text{ A}\to \text{D}$
• Utilize union axiom on $\text{ A}\to \text{C}$ and $\text{ A}\to \text{D}$, to get $\text{ A}\to \text{CD}$
• The transitive axiom is used on the provided functional dependency $\text{ CD}\to \text{E}$ and also with the functional dependency $\text{ A}\to$