Chapter 7, Problem 18PE

Explanation of Solution

To prove:

The new and original definitions of $3\text{NF}$ are equal.

Proof:

• Consider the new definition of a schema given in the question in third Normal Form $\left(3\text{NF}\right)$ along with the original one given in Section $7$ $3$ $2$ of the textbook.
• As per the definition given in Section $7$ $3$ $2$ of the textbook, consider a schema $\text{R}$ which is in $3\text{NF}$.
• By contradiction, this schema can be proved that this is in $3\text{NF}$ as per the new definition.
• Let “$\text{A}$” be the non-primitive attribute in schema and it is transitively dependent on a key $\text{α}$ for the schema.
• The above statement represents $\square \text{β}$ is a subset of $\text{R}$ such that $\text{β}\to \text{A}$ $\text{α}\to \text{β}$ when     $\text{A}\notin \text{α}$, $\text{A}\notin \text{β}$ and $\text{β}\to \text{A}$ is not true.
• If this is the situation, the original definition of $3\text{NF}$ is violated the functional dependency as,
  • $\text{A}\notin \text{β}$ represents that $\text{β}\to \text{A}$ is a non-trivial functional dependency.
  • $\text{B}$ is not a super key as $\text{β}\to \text{α}$ is not true.
  • As an attribute “$\text{A}$” is non-prime, it is not a candidate key.
• As per the original definition given in Section $7$ $3$ $2$, the schema $\text{R}$ can be proved that this is in $3\text{NF}$