7.19

A functional dependency $\alpha \to \beta$ is called a partial dependency if there is a proper subset $\gamma$ of $\alpha$ such that $\gamma \to \beta$; we say that $\beta$ is partially dependent on $\alpha$. A relation schema $R$ is in second normal form (2NF) if each attribute $A$ in $R$ meets one of the following criteria:

• It appears in a candidate key.
• It is not partially dependent on a candidate key.

Show that every 3NF schema is in 2NF. (Hint: Show that every partial dependency is a transitive dependency.)

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Referring to the definitions in Exercise 7.18, a relation schema $R$ is said to be in 3NF if there is no nonprime attribute $A$ in $R$ for which $A$ is transitively dependent on a key for $R$.

We can also rewrite the definition of 2NF given here as:

“A relation schema $R$ is in 2NF if no nonprime attribute $A$ is partially dependent on any candidate key for $R$.”

To prove that every 3NF schema is in 2NF, it suffices to show that if a nonprime attribute $A$ is partially dependent on a candidate key $\alpha$, then $A$ is also transitively dependent on the key $\alpha$.

Let $A$ be a nonprime attribute in $R$. Let $\alpha$ be a candidate key for $R$. Suppose $A$ is partially dependent on $\alpha$.

• From the definition of a partial dependency, we know that for some proper subset $\beta$ of $\alpha$, $\beta \to A$.

• Since $\beta \subset \alpha ,\alpha \to \beta$. Also, $\beta \to \alpha$ does not hold, since $\alpha$ is a candidate key.

• Finally, since $A$ is nonprime, it cannot be in either $\beta$ or $\alpha$.

Thus we conclude that $\alpha \to A$ is a transitive dependency. Hence we have proved that every 3NF schema is also in 2NF.