Consider an Entity-Relationship (ER) model in which entity sets $E_1$ and $E_2$ are connected by an $m:n$ relationship $R_{12}$. $E_1$ and $E_3$ are connected by a $1:n$ (1 on the side of $E_1$ and $n$ on the side of $E_3$) relationship $R_{13}$. $E_1$ has two single-valued attributes $a_{11}$ and $a_{12}$ of which $a_{11}$ is the key attribute. $E_2$ has two single-valued attributes $a_{21}$ and $a_{22}$ of which $a_{21}$ is the key attribute. $E_3$ has two single-valued attributes $a_{31}$ and $a_{32}$ of which $a_{31}$ is the key attribute. The relationships do not have any attributes. If a relational model is derived from the above ER model, then the minimum number of relations that would be generated if all the relations are in 3NF is _______.

To convert the given ER diagram to a relational model with minimal tables in 3NF:

1. Entity Set $E_1$:

   • Has attributes $a_{11}$ (primary key) and $a_{12}$.

   • Generates relation: $T_1(\underline{a_{11}}, a_{12})$.

2. Entity Set $E_2$:

   • Has attributes $a_{21}$ (primary key) and $a_{22}$.

   • Generates relation: $T_2(\underline{a_{21}}, a_{22})$.

3. Relationship $R_{12}$ ($m:n$ between $E_1$ and $E_2$):

   • An $m:n$ relationship cannot be merged into either entity table without violating 1NF/introducing NULL values or redundancy.

   • Generates a separate relation with the composite primary key: $T_{12}(\underline{a_{11}, a_{21}})$.

4. Relationship $R_{13}$ ($1:n$ from $E_1$ to $E_3$) and Entity Set $E_3$:

   • Since each entity in $E_3$ is related to at most one entity in $E_1$, the relationship $R_{13}$ can be merged into the entity relation of $E_3$ by including the primary key of $E_1$ ($a_{11}$) as a foreign key.

   • Generates relation: $T_3(\underline{a_{31}}, a_{32}, a_{11})$.

Thus, the minimum number of relations required is 4 [p. 17, Q. 48].