Representing Many-to-One Relationships with Functional Dependencies

In database design, understanding relationship types between entity sets is crucial. A many-to-one relationship between two entity sets, say r₁ and r₂, implies that multiple instances of entity r₁ can be associated with a single instance of entity r₂, but an instance of r₂ can be associated with zero or more instances of r₁. The relationship direction matters. If it's "many-to-one from r₁ to r₂", it means many r₁'s relate to one r₂.

Functional dependencies are a way to describe constraints on data in a relational database. A functional dependency X → Y means that if two tuples have the same values for the attributes in set X, they must also have the same values for the attributes in set Y. Essentially, X uniquely determines Y.

We are given entity sets r₁ and r₂, with primary keys Pk(r₁) and Pk(r₂) respectively. We need to represent a many-to-one relationship that exists from r₁ to r₂ using functional dependencies involving their primary keys.

Consider the relationship: "Many instances of r₁ relate to one instance of r₂". This means if you take any specific instance of r₁, it is related to exactly one instance of r₂. Therefore, knowing the primary key of an r₁ instance (Pk(r₁)) should uniquely identify the primary key of the related r₂ instance (Pk(r₂)). This directly translates to a functional dependency:

\begin{equation*} \text{Pk(r_1)} \rightarrow \text{Pk(r_2)} \end{equation*}

Let's look at why the other options do not correctly represent a many-to-one relationship from r₁ to r₂:

• \(\text{Pk(r_2)} \rightarrow \text{Pk(r_1)}\)

  This would mean that knowing the primary key of an r₂ instance uniquely identifies the primary key of a related r₁ instance. This describes a one-to-many relationship from r₂ to r₁ (or equivalently, a many-to-one from r₁ to r₂ *only if* every r₂ instance is related to exactly one r₁ instance, which is a special case or a one-to-one). However, in a general many-to-one from r₁ to r₂, an r₂ instance can be related to *many* r₁ instances, so Pk(r₂) does not determine Pk(r₁).

• \(\text{Pk(r_2)} \rightarrow \text{Pk(r_1)} \text{ and } \text{Pk(r_1)} \rightarrow \text{Pk(r_2)}\)

  This combination of dependencies means that Pk(r₁) uniquely determines Pk(r₂), and Pk(r₂) uniquely determines Pk(r₁). This is the definition of a one-to-one relationship between the two entity sets.

• \(\text{Pk(r_2)} \rightarrow \text{Pk(r_1)} \text{ or } \text{Pk(r_1)} \rightarrow \text{Pk(r_2)}\)

  This represents either a one-to-many from r₂ to r₁ or a many-to-one from r₁ to r₂ (or a one-to-one if both hold). It is not a precise representation *specifically* for the many-to-one relationship from r₁ to r₂ without the possibility of the other case holding exclusively.

Therefore, the functional dependency that represents a many-to-one relationship from r₁ to r₂, where many r₁ instances relate to one r₂ instance, is Pk(r₁) → Pk(r₂).

Revision Table: Key Concepts

Additional Information: Representing Relationships in Relational Model

When translating an Entity-Relationship (ER) model into a relational database schema, different relationship types are represented using primary and foreign keys.

• One-to-Many Relationship (e.g., Department 1 : * Employee): The primary key of the "one" side (Department ID) is included as a foreign key in the table on the "many" side (Employee table). This foreign key in the Employee table would functionally determine the Department ID, aligning with Pk(Many) → Pk(One).
• Many-to-One Relationship (e.g., Employee * : 1 Department): This is the same as the one-to-many relationship described above, just viewed from the other direction. The primary key of the "one" side (Department ID) is included as a foreign key in the table on the "many" side (Employee table). The functional dependency is Pk(Many) → Pk(One), which in the context of the question is Pk(r₁) → Pk(r₂), where r₁ is the 'many' side (Employee) and r₂ is the 'one' side (Department).
• One-to-One Relationship (e.g., Employee 1 : 1 Parking Spot): The primary key of one table can be included as a foreign key in the other table. If both tables are mandatory participation, sometimes they are merged. The functional dependencies Pk(Table1) → Pk(Table2) and Pk(Table2) → Pk(Table1) both hold.
• Many-to-Many Relationship (e.g., Student * : * Course): A new linking table is created. This linking table contains the primary keys of both original tables as foreign keys. The primary key of the linking table is typically a composite key formed by these two foreign keys. No functional dependency exists between the primary keys of the original two tables alone (Pk(r₁) does not determine Pk(r₂), and Pk(r₂) does not determine Pk(r₁)).

Understanding how relationship cardinality translates to functional dependencies between primary keys is fundamental for correct database design and normalization.