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Chapter 10: Problem 7

What role do Armstrong's inference rules- -the three inference rules IR 1 through IR3-play in the development of the theory of relational design?

Short Answer

Expert verified
Armstrong's inference rules, consisting of reflexivity, augmentation, and transitivity, play a key role in relational database design theory. They form the foundation of functional dependency theory used in normalization processes to reduce data redundancy and improve data integrity in a relational database.

Step by step solution

01

Understanding Armstrong's Inference Rules

Armstrong's inference rules consist of three basic rules: reflexivity, augmentation, and transitivity. They serve as the fundamental principles for inferring all the possible functional dependencies on a relational database.
02

Significance of Armstrong's Rules in Relational Database Design Theory

These rules form the basis of formalizing the properties of functional dependencies and normalization of relational schema. They are important for proving the correctness of functional dependencies and ensuring data integrity in a relational database.
03

Role in Data Normalization

Armstrong's Inference Rules are specifically essential during the normalization process. Normalization seeks to reduce data redundancy and improve data integrity. By systematically applying these rules, database designers can effectively decompose relations while preserving necessary dependencies, leading to an efficient, well-designed relational database schema.

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Most popular questions from this chapter

Prove or disprove the following inference rules for functional dependencies. A proof can be made either by a proof argument or by using inference rules IR1 through IR3. A disproof should be performed by demonstrating a relation instance that satisfies the conditions and functional dependencies in the left-hand side of the inference rule but does not satisfy the dependencies in the right-hand side. a. \(\\{W \rightarrow Y, X \rightarrow Z\\} \vDash\\{W X \rightarrow Y\\}\) b. \(\\{X \rightarrow Y\\}\) and \(Y \supseteq Z \vDash\\{X \rightarrow Z\\}\) ?. \(\\{X \rightarrow Y, X \rightarrow W, W Y \rightarrow Z\\} \vDash\\{X \rightarrow Z\\}\) d. \(\\{X Y \rightarrow Z, Y \rightarrow W\\} \vDash\\{X W \rightarrow Z\\}\) e. \(\\{X \rightarrow Z, Y \rightarrow Z\\} \vDash\\{X \rightarrow Y\\}\) f. \(\quad\\{X \rightarrow Y, X Y \rightarrow Z\\} \vDash\\{X \rightarrow Z\\}\) \(\mathrm{g} .\\{X \rightarrow Y, Z \rightarrow W\\} \vDash\\{X Z \rightarrow Y W\\}\) h. \(\\{X Y \rightarrow Z, Z \rightarrow X\\} \vDash\\{Z \rightarrow Y\\}\) ¡. \(\\{X \rightarrow Y, Y \rightarrow Z\\} \vDash\\{X \rightarrow Y Z\\}\) j. \(\quad\\{X Y \rightarrow Z, Z \rightarrow W\\} \vDash\\{X \rightarrow W\\}\)

, Odate, Cust#, Total_amount) ORDER- ITEM(O#, I#, Qty_order… # Consider the following relations for an order-processing application database at \(\mathrm{ABC},\) Inc. ORDER (O#, Odate, Cust#, Total_amount) ORDER-ITEM(O#, I#, Qty_ordered, Total_price, Discount\%) Assume that each item has a different discount. The Total_PRICE refers to one item, OOATE is the date on which the order was placed, and the Total_AMOUNT is the amount of the order. If we apply a natural join on the relations ORDER-ITEM and ORDER in this database, what does the resulting relation schema look like? What will be its key? Show the FDs in this resulting relation. Is it in \(2 \mathrm{NF}\) ? Is it in \(3 \mathrm{NF}\) ? Why or why not? (State assumptions, if you make any.)

What undesirable dependencies are avoided when a relation is in \(2 \mathrm{NF}\) ?

What is meant by the closure of a set of functional dependencies? Illustrate with an example.

} \\ \hline & & & \\ 10 & \mathrm{b} 1 & \mathrm{c} 1 & \\# 1… # Consider the following relation: $$\begin{array}{llll} \mathrm{A} & \mathbf{B} & \mathbf{C} & \text { TUPLE# } \\ \hline & & & \\ 10 & \mathrm{b} 1 & \mathrm{c} 1 & \\# 1 \\ 10 & \mathrm{b} 2 & \mathrm{c} 2 & \\# 2 \\ 11 & \mathrm{b} 4 & \mathrm{c} 1 & \\# 3 \\ 12 & \mathrm{b} 3 & \mathrm{c} 4 & \\# 4 \\ 13 & \mathrm{b} 1 & \mathrm{c} 1 & \\# 5 \\ 14 & \mathrm{b} 3 & \mathrm{c} 4 & \\# 6 \end{array}$$ a. Given the previous extension (state), which of the following dependencies may hold in the above relation? If the dependency cannot hold, explain why by specifying the tuples that cause the violation. i. \(A \rightarrow B,\) ii. \(B \rightarrow C,\) iii. \(C \rightarrow B,\) iv. \(B \rightarrow A, v . C \rightarrow A\) b. Does the above relation have a potential candidate key? If it does, what is it? If it does not, why not?
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