Show the decomposition process for better understanding your solutions.
1. Design by Decomposition
Consider the following relational schema:
Sale(salesman, store, commission_rate, city, date_sold, product, style, color) // a salesman sold a product on a particular day
Product(product, style, color, price) // prices, available styles and colors for products
Make the following assumptions, and only these assumptions, about the real world being modeled:
– Each salesman works in one store and has one fixed commission rate.
– Each store is in one city.
– A given product always has the same price, regardless of style or color.
– Each product is available in one or more styles and one or more colors, and each product is available in all combinations of styles and colors for that product.
Sale does not contain duplicates: If a salesman sells more than one of a given product in a given style and color on a given day, still only one tuple appears in relation Sale to record that fact.
(a) Specify a set of completely nontrivial functional dependencies for relations Sale and Product that encodes the assumptions described above and no additional assumptions.
(b) Based on your functional dependencies in part (a), specify all minimal keys for relations Sale and Product.
(c) Is the schema in Boyce-Codd Normal Form (BCNF) according to your answers to (a) and (b)? If not, give a decomposition into BCNF.
(d) Now consider your decomposed relations from part (c), or the original relations if you did not need to decompose them for part (c). Specify a set of nontrivial multivalued dependencies for relations Sale and Product that encodes the assumptions described above and no additional assumptions. Do not include multivalued dependencies that also are functional dependencies.
(e) Are the relations you used in part (d) in Fourth Normal Form (4NF) according to your answers for (a)-(d)? If not, give a decomposition into 4NF.

 

 

Sample solution

Dante Alighieri played a critical role in the literature world through his poem Divine Comedy that was written in the 14th century. The poem contains Inferno, Purgatorio, and Paradiso. The Inferno is a description of the nine circles of torment that are found on the earth. It depicts the realms of the people that have gone against the spiritual values and who, instead, have chosen bestial appetite, violence, or fraud and malice. The nine circles of hell are limbo, lust, gluttony, greed and wrath. Others are heresy, violence, fraud, and treachery. The purpose of this paper is to examine the Dante’s Inferno in the perspective of its portrayal of God’s image and the justification of hell. 

In this epic poem, God is portrayed as a super being guilty of multiple weaknesses including being egotistic, unjust, and hypocritical. Dante, in this poem, depicts God as being more human than divine by challenging God’s omnipotence. Additionally, the manner in which Dante describes Hell is in full contradiction to the morals of God as written in the Bible. When god arranges Hell to flatter Himself, He commits egotism, a sin that is common among human beings (Cheney, 2016). The weakness is depicted in Limbo and on the Gate of Hell where, for instance, God sends those who do not worship Him to Hell. This implies that failure to worship Him is a sin.

God is also depicted as lacking justice in His actions thus removing the godly image. The injustice is portrayed by the manner in which the sodomites and opportunists are treated. The opportunists are subjected to banner chasing in their lives after death followed by being stung by insects and maggots. They are known to having done neither good nor bad during their lifetimes and, therefore, justice could have demanded that they be granted a neutral punishment having lived a neutral life. The sodomites are also punished unfairly by God when Brunetto Lattini is condemned to hell despite being a good leader (Babor, T. F., McGovern, T., & Robaina, K. (2017). While he commited sodomy, God chooses to ignore all the other good deeds that Brunetto did.

Finally, God is also portrayed as being hypocritical in His actions, a sin that further diminishes His godliness and makes Him more human. A case in point is when God condemns the sin of egotism and goes ahead to commit it repeatedly. Proverbs 29:23 states that “arrogance will bring your downfall, but if you are humble, you will be respected.” When Slattery condemns Dante’s human state as being weak, doubtful, and limited, he is proving God’s hypocrisy because He is also human (Verdicchio, 2015). The actions of God in Hell as portrayed by Dante are inconsistent with the Biblical literature. Both Dante and God are prone to making mistakes, something common among human beings thus making God more human.

To wrap it up, Dante portrays God is more human since He commits the same sins that humans commit: egotism, hypocrisy, and injustice. Hell is justified as being a destination for victims of the mistakes committed by God. The Hell is presented as being a totally different place as compared to what is written about it in the Bible. As a result, reading through the text gives an image of God who is prone to the very mistakes common to humans thus ripping Him off His lofty status of divine and, instead, making Him a mere human. Whether or not Dante did it intentionally is subject to debate but one thing is clear in the poem: the misconstrued notion of God is revealed to future generations.

 

References

Babor, T. F., McGovern, T., & Robaina, K. (2017). Dante’s inferno: Seven deadly sins in scientific publishing and how to avoid them. Addiction Science: A Guide for the Perplexed, 267.

Cheney, L. D. G. (2016). Illustrations for Dante’s Inferno: A Comparative Study of Sandro Botticelli, Giovanni Stradano, and Federico Zuccaro. Cultural and Religious Studies4(8), 487.

Verdicchio, M. (2015). Irony and Desire in Dante’s” Inferno” 27. Italica, 285-297.

Sample Answer

Sample Answer

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.

 

 

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