503 Inference Rules for Functional Dependencies PDF

503 Inference Rules for Functional == == Dependencies ** Inference Rules, also known as Armstrong’s Axioms, are foundational rules used to derive ** ** ** a...

503 Inference Rules for Functional == == Dependencies ** Inference Rules, also known as Armstrong’s Axioms, are foundational rules used to derive ** ** ** all possible functional dependencies from a given set of dependencies in a relational ** ** database. These rules—Reflexivity, Augmentation, and Transitivity—are essential for understanding ** ** ** ** ** ** and organizing functional dependencies effectively, which aids in database normalization ** ** and minimizes redundancy. Let’s go through these three main rules with examples and tables based on a Student ` ` database. 1. Reflexivity Rule *~ ~* The Reflexivity Rule states that if a set of attributes Y is a subset of a set of attributes X , ** ** ` ` *** *** ` ` then X → Y holds. ` ` ==* This means an attribute or a combination of attributes always determines itself or any ** subset of itself. ** *== ** Notation: ** * If Y ⊆ X , then X → Y. * ` ` * * ` ` ** Example: Consider the following Student table with attributes {Student_ID, Name, ** ` ` ` Course}. ` ` Student_ID ` Name Course 101 Alice Smith Math 102 Bob Johnson Science 103 Carol White History In this table: ` {Student_ID, Name} → Student_ID ` * is a valid dependency * * because Student_ID is a subset of {Student_ID, Name}. ` ` ** ** ` ` * ` {Student_ID, Course} → Course ` * also holds by reflexivity, * * as Course is a subset of {Student_ID, Course}. ` ` ** ** ` ` * The reflexivity rule emphasizes that any attribute set inherently determines itself and any ** ** * subset of its components. *  ❗ 2. Augmentation Rule *~ ~* The Augmentation Rule states that if X → Y holds, then adding an attribute Z to both ** ** ` ` * == ` ` sides of the dependency will still result in a valid dependency. == * * == This rule allows attributes to be added to a dependency without altering its validity. == * ** Notation: ** * If X → Y , ` ` then XZ → YZ ` ` (where XZ is the union of X and Z ). ` ` ` ` ` ` * *** Example: ** Using the same Student table above, suppose we know that Student_ID → Name ` ` ` ` holds true, meaning that each Student_ID uniquely identifies a Name. ` ` ` ` * * Now, let’s add Course to both sides: ` ` * * ` {Student_ID, Course} → {Name, Course} ` also holds by the augmentation rule. * * If we know that Student_ID determines Name , then {Student_ID, Course} will ` ` ` ` ` ` still uniquely determine both Name and Course. ` ` ` ` * * This rule is especially useful when constructing complex dependencies that involve multiple attributes while ensuring no loss of information. * 3. Transitivity Rule *~ ~* The Transitivity Rule is similar to the transitive property in mathematics. ** ** * It states that if X → Y and Y → Z hold, then X → Z is also a valid dependency. ` ` ` ` ` ` * * Transitivity is crucial for identifying indirect dependencies that can contribute to ** ** redundancy if not properly managed. * ** Notation: **  If X → Y and Y → Z , ` ` ` ` then X → Z. ` ` ** Example: ** Consider an expanded Student_Department table with the following attributes: ` ` Student_ID Department_ID Department_Name 101 D01 Science 102 D02 Arts 103 D03 Commerce In this table, suppose the following dependencies hold: ` Student_ID → Department_ID : ` Each student is assigned a unique department. ` Department_ID → Department_Name : ` Each department ID uniquely identifies a department name. * By the transitivity rule: * * ` Student_ID → Department_Name can be inferred, meaning Student_ID indirectly ` ` ` determines Department_Name through Department_ID. ` ` ` ` * This transitive dependency highlights the need to normalize data to reduce redundancy, ** ** * as it indicates that storing Department_Name with Student_ID could result in unnecessary ` ` ` ` data duplication. * Summary Table of Inference Rules == Rule == == Definition == == Example == ** Reflexivity ** If Y is a subset of X, then X → ` ` ` ` ` ` {Student_ID, Name} → Y ` Student_ID ` ** Augmentation ** If X → Y , ` ` If Student_ID → Name , ` ` then XZ → YZ ` ` then {Student_ID, Course} ` → {Name, Course} ` ** Transitivity ** If X → Y and Y → Z , then X → ` ` ` ` ` If Student_ID → ` Z ` Department_ID and ` ` Department_ID → Department_Name , ` then Student_ID → ` Department_Name `