$1.$ Entities:

Entities are objects or concepts that you want to store information about in your database. In this database, we have four entities:

Student: This entity stores information about students, including their unique ID$,$ name, and major.

Course: This entity stores information about courses, including their unique ID$,$ name, and credit hours.

Enrollment: This entity stores information about student enrollments in courses, including a unique enrollment ID$,$ the ID of the student, the ID of the course, and the grade earned by the student in the course.

Instructor: This entity stores information about instructors, including their unique ID$,$ name, and department.

Each entity has a primary key $($PK$),$ which is a unique identifier for each record in the entity.

$2.$ Relationships:

Relationships describe how the entities in the database are related to each other. In this database, we have two types of relationships:

Disjoint relationship: Course can be required or elective for a major. This relationship can be modeled using a disjoint subtype entity. The subtype entity can be named "Course$\_$Type" with the following attributes:

course$\_$type$\_$id $($PK$)$

course$\_$id $($FK to Course$)$

type $($Required or Elective$)$

Overlap relationship: A student can take multiple courses, and an instructor can teach multiple courses. This relationship can be modeled using a Many$-$to$-$Many $($M:N$)$ relationship between Student and Course, and between Instructor and Course. The Enrollment entity can act as a junction table with foreign keys to both Student and Course entities. Similarly, a junction table named "Teaching" can be created with foreign keys to Instructor and Course entities.

$3.$ Union:

The Student and Instructor entities can be merged using a union. Both entities have common attributes such as name and department. The resulting entity can be named "Person" with the following attributes:

person$\_$id $($PK$)$

name

department

type $($Student or Instructor$)$

Explanation:

A union is a way to combine two or more entities that have similar attributes. In this database, the Student and Instructor entities have similar attributes $($name and department$).$ To combine them into a single entity, we create a union called Person, which has a unique ID$,$ name, department, and an attribute called "type" to indicate whether the record is a student or an instructor.

$4.$ Relationships:

We have three types of relationships in this database:

$1$:$1$ relationship: This is a relationship where each record in one entity is related to only one record in the other entity. In this database, the Student entity has a $1$:$1$ relationship with the Major entity. This is because each student can have only one major. To model this relationship, we add a foreign key called "major$\_$id$"$ to the Student entity, which references the Major entity.

$1$:N relationship: This is a relationship where each record in one entity can be related to multiple records in the other entity, but each record in the other entity can be related to only one record in the first entity. In this database, the Course entity has a $1$:N relationship with the Instructor entity. This is because a course can only have one instructor, but an instructor can teach multiple courses. To model this relationship, we add a foreign key called "instructor$\_$id$"$ to the Course entity, which references the Instructor entity.

N:M relationship: This is a relationship where each record in one entity can be related to multiple records in the other entity, and vice versa. In this database, the Student entity has an N:M relationship with the Course entity