To better understand the decomposition process for the given relational schema, we will follow the steps outlined in the prompt. We will analyze the schema, identify functional dependencies, determine keys, evaluate normalization forms, and identify multivalued dependencies.
(a) Functional Dependencies
Given the assumptions, we can derive the following nontrivial functional dependencies for the relations Sale and Product:
For Sale:
1. salesman → store, commission_rate, city
(Each salesman works in one store and has one fixed commission rate.)
2. store → city
(Each store is in one city.)
3. product, style, color → price
(A given product always has the same price, regardless of style or color.)
4. salesman, store, date_sold, product, style, color → commission_rate
(The combination of these attributes uniquely identifies a sale with its associated commission rate.)
For Product:
5. product → price
(Each product has a consistent price regardless of style and color.)
6. product → style, color
(Each product is available in multiple styles and colors.)
(b) Minimal Keys
Based on the functional dependencies identified:
For Sale:
To determine a minimal key for Sale, we need a combination of attributes that can uniquely identify each tuple:
– The attributes salesman, store, date_sold, product, style, color together form a candidate key since they can identify each sale uniquely.
Thus, the minimal key for relation Sale is:
– {salesman, store, date_sold, product, style, color}
For Product:
The minimal key for Product is simply:
– {product}
(c) Boyce-Codd Normal Form (BCNF)
A relation is in BCNF if every non-trivial functional dependency (X → Y) holds that (X) is a superkey.
Evaluating BCNF for Sale:
– The functional dependency salesman → store, commission_rate, city implies that salesman is not a superkey since it does not uniquely identify all attributes in Sale. Therefore, Sale is not in BCNF.
Decomposition for Sale:
To decompose Sale into BCNF, we can create two new relations:
1. Salesman_Info(salesman, store, commission_rate, city)
This relation captures information related to the salesman.
2. Sale_Details(salesman, date_sold, product, style, color)
This relation captures the sales information along with the date sold.
Now the functional dependencies can be maintained without violating BCNF.
(d) Multivalued Dependencies
Next, we identify nontrivial multivalued dependencies for both relations based on the assumptions:
For Sale_Details:
1. product ↠ style
(A product can have multiple styles.)
2. product ↠ color
(A product can have multiple colors.)
For Salesman_Info:
No multivalued dependencies are present since each salesman relates directly to a single set of information (store and commission rate).
(e) Fourth Normal Form (4NF)
A relation is in 4NF if it is in BCNF and has no nontrivial multivalued dependencies that are not functional dependencies.
Evaluating 4NF for Sale_Details:
The relation Sale_Details has multivalued dependencies (product ↠ style and product ↠ color) that are not functional dependencies. Hence it is not in 4NF.
Decomposition for Sale_Details:
To convert Sale_Details into 4NF, we can decompose it into two relations:
1. Sale_Styles(product, style)
This captures the relationship between products and their available styles.
2. Sale_Colors(product, color)
This captures the relationship between products and their available colors.
Summary of Decompositions
1. Original Relation:
– Sale(salesman, store, commission_rate, city, date_sold, product, style, color)
– Product(product, style, color, price)
2. After Step (c) Decomposition:
– Salesman_Info(salesman, store, commission_rate, city)
– Sale_Details(salesman, date_sold, product, style, color)
3. After Step (e) Decomposition of Sale_Details:
– Salesman_Info remains unchanged.
– Sale_Styles(product, style)
– Sale_Colors(product, color)
These decompositions ensure that all relations comply with BCNF and 4NF standards while maintaining data integrity according to the specified assumptions.
