Consider two entity sets E1(A11, A12, A13) and E2(A21,A22,A23) with A11 and A21 as PK.

A22 is a multivalued attribute. R12 is a many to many relationship with total participation on both side. What is the min number of relations required to convert this ER model to relational model in 3NF?

The correct answer is 4.

To determine the minimum number of relations (tables) required to represent the given Entity-Relationship (ER) model in the relational model in Third Normal Form (3NF), we analyze each component of the ER diagram step-by-step:
1. Entity E1: E1 has attributes A11, A12, and A13, with A11 as the Primary Key (PK). This entity is represented by a relation:
E1(A11, A12, A13)
This gives us 1 relation.

2. Entity E2 and its multivalued attribute: E2 has attributes A21, A22, and A23, with A21 as the PK. However, A22 is a multivalued attribute. In relational database design, multivalued attributes cannot be stored directly in the main entity table because doing so would violate First Normal Form (1NF). Therefore, they must be split into a separate relation:
- The main relation for E2: E2(A21, A23)
- The relation for the multivalued attribute A22: E2_A22(A21, A22), where both attributes together form the composite primary key.
This gives us 2 relations for E2.

3. Relationship R12: R12 is a many-to-many relationship between E1 and E2. A many-to-many relationship always requires its own separate junction table containing the primary keys of the participating entities as foreign keys. Even with total participation on both sides, a many-to-many relationship cannot be merged into either E1 or E2 without causing redundancy or violating normal forms. Thus, we need a separate relation for the relationship:
R12(A11, A21)
This gives us 1 relation for the relationship.

Summing up the required relations:
- Relation for E1: 1
- Relations for E2 (including multivalued attribute A22): 2
- Relation for many-to-many relationship R12: 1

Total minimum number of relations required = 1 + 2 + 1 = 4.