Chapter 7, Problem 25E

Explanation of Solution

To show the Armstrong’s reflexivity axiom is found:

• A functional dependency $\text{A}\to \text{B}$ which holds true only when for all pairs of records t1 and t2 in a relation r(R) such that t1[A]=t2[A] then t1[B]=t2[B] also holds true.
• The Armstrong’s reflexivity axiom defines that if “A” is a set of attributes and “B” is a subset of “A”, then $\text{A}\to \text{B}$.
• Assume ꓱt1,t2 such that t1[A]=t2[A] to prove the soundness of the axiom.
• If “B” is a subset of “A”, then t1[B]=t2[B].
• From the functional dependency definitions with t1[A]=t2[A] and t1[B]=t2[B], the user have $\text{A}\to \text{B}$.

To prove the Armstrong’s augmentation axiom is found:

• This axiom defines that If “C” is a set of attributes and $\text{A}\to \text{B}$, the $CA\to CB$ also holds.
• Assume ꓱt1,t2 such that t1[CA]=t2[CA] to prove the soundness of the augmentation  axiom.
• If “C” is a subset of “CA”, then t1[C]=t2[C].
• Similarly if “A” is a subset of “AC”, then t1[A]=t2[A].
• Now, the user can have t1[B]=t2[B] from the definition of $\text{A}\to \text{B}$