GATE CSE 1987 Question 2n: Mastering Functional Dependencies in Databases

Are you preparing for GATE CSE and struggling with database concepts? Let’s dive deep into a classic question from GATE CSE 1987 that tests your understanding of functional dependencies. This comprehensive guide will not only help you solve the problem but also strengthen your grasp on this crucial database concept.

Original GATE Question

Year: 1987

Question Number: 2n

Question: Does a relation r with schema (X, Y) satisfy the functional dependency X → Y, and can the tuples ⟨1, 2⟩ and ⟨2, 2⟩ both be in r simultaneously?

Detailed Solution

To tackle this question, we need to understand functional dependencies thoroughly and apply our knowledge to the given scenario.

Understanding Functional Dependencies

A functional dependency X → Y means that the attribute (or set of attributes) X uniquely determines the attribute (or set of attributes) Y. In simpler terms, if we know the value of X, we can determine the value of Y.

Key Rules of Functional Dependencies

• Uniqueness Rule: If all values of X are unique, X → Y always holds, regardless of Y values.
• Sameness Rule: If all values of Y are the same, X → Y always holds, regardless of X values.
• Consistency Rule: For X → Y to hold, if two tuples have the same X value, they must have the same Y value.
• Violation Rule: A violation occurs only when two or more tuples have the same X value but different Y values.

Analyzing the Given Scenario

Let’s apply these rules to our relation r with tuples ⟨1, 2⟩ and ⟨2, 2⟩:

• X values are 1 and 2 (different)
• Y values are both 2 (same)

Applying the Rules

1. Uniqueness Rule: The X values (1 and 2) are unique. This satisfies the functional dependency X → Y.
2. Sameness Rule: All Y values are the same (2). This also satisfies X → Y.
3. Consistency Rule: There are no repeated X values, so this rule is not violated.
4. Violation Rule: There’s no violation as we don’t have any repeated X values with different Y values.

Conclusion

Yes, the relation r with schema (X, Y) satisfies the functional dependency X → Y, and the tuples ⟨1, 2⟩ and ⟨2, 2⟩ can indeed be in r simultaneously.

Solution Summary

• The functional dependency X → Y holds in this relation due to:
  • Unique X values (1 and 2)
  • Identical Y values (both 2)
• The given tuples ⟨1, 2⟩ and ⟨2, 2⟩ can coexist without violating the functional dependency.
• This scenario satisfies both the Uniqueness Rule and the Sameness Rule of functional dependencies.

Common Student Doubts (FAQs)

Q1: Why does X → Y hold when X values are different?

A: The functional dependency X → Y means that each X value should correspond to only one Y value. When X values are different, they can have any Y values without violating this rule. The dependency is only violated if the same X value has different Y values.

Q2: Does the order of tuples matter in functional dependencies?

A: No, the order of tuples is irrelevant for functional dependencies. What matters is the relationship between X and Y values across all tuples in the relation.

Q3: Would the dependency still hold if we had a tuple ⟨1, 3⟩ instead of ⟨1, 2⟩?

A: Yes, the functional dependency would still hold even with ⟨1, 3⟩ instead of ⟨1, 2⟩. The key is that the X values (1 and 2) remain unique, satisfying the Uniqueness Rule of functional dependencies.

Q4: Can we have more tuples in this relation while maintaining the functional dependency?

A: Yes, we can add more tuples as long as they don’t violate the rules. For example, we could add ⟨3, 4⟩ or ⟨4, 2⟩, and the functional dependency would still hold.

Tutorial: Mastering Functional Dependencies

To fully grasp functional dependencies, focus on these key areas:

• Relational Database Fundamentals: Understanding tables, attributes, and tuples; primary and foreign keys
• Functional Dependency Basics: Definition and notation; identifying dependencies in a relation
• Types of Functional Dependencies: Trivial and non-trivial dependencies; full and partial dependencies
• Armstrong’s Axioms: Reflexivity, augmentation, and transitivity; applying these rules to derive new dependencies
• Normalization: First Normal Form (1NF), Second Normal Form (2NF), Third Normal Form (3NF), Boyce-Codd Normal Form (BCNF)
• Dependency Preservation and Lossless Join: Ensuring data integrity during normalization

Practice Example

Consider this table:

Identify the functional dependencies:

• StudentID → Course (Not valid, as 101 takes multiple courses)
• Course → Instructor (Not valid, as Math is taught by different instructors)
• {StudentID, Course} → Instructor (Valid)

Remember, mastering functional dependencies is crucial for efficient database design and management. Keep practicing with various examples to solidify your understanding!