GATE CSE 1987 Question 2n: Functional Dependencies in Databases
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 answer this question, we need to understand the concept of functional dependencies and apply it to the given scenario.
Understanding Functional Dependencies
A functional dependency X → Y means that the value of X uniquely determines the value of Y. In other words, if we know X, we can determine Y.
Rules of Functional Dependencies
- Rule 1: If all values of X are unique, X → Y always holds, regardless of Y values.
- Rule 2: If all values of Y are the same, X → Y always holds, regardless of X values.
- Rule 3: For X → Y to hold, if two tuples have the same X value, they must have the same Y value.
- Rule 4: A violation occurs only when two or more tuples have the same X value but different Y values.
Applying to the Given Scenario
In the given relation r with tuples ⟨1, 2⟩ and ⟨2, 2⟩:
- X values are 1 and 2 (different)
- Y values are both 2 (same)
According to Rule 1, since the X values (1 and 2) are unique, the functional dependency X → Y holds regardless of the Y values.
Additionally, according to Rule 2, since all Y values are the same (2), the functional dependency X → Y also holds regardless of the X values.
There is no violation of Rule 3 or Rule 4 because 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 both be in r simultaneously.
Solution Summary
- The functional dependency X → Y holds in this relation.
- Unique X values (1 and 2) satisfy the dependency regardless of Y values.
- Identical Y values (both 2) also satisfy the dependency regardless of X values.
- The given tuples ⟨1, 2⟩ and ⟨2, 2⟩ can coexist without violating the functional dependency.
Common Doubts and Solutions (FAQs)
Q1: Why does X → Y hold when X values are different?
A: The functional dependency X → Y means that for any given X value, there should be only one corresponding 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 doesn’t matter for functional dependencies. What matters is the relationship between X and Y values across all tuples in the relation.
Q3: How would this change if we had a tuple ⟨1, 3⟩ instead of ⟨1, 2⟩?
A: Even if we had ⟨1, 3⟩ instead of ⟨1, 2⟩, the functional dependency would still hold. The key is that the X values (1 and 2) remain unique, satisfying Rule 1 of functional dependencies.
Tutorial: Understanding Functional Dependencies
To fully grasp the concept of functional dependencies, students should study the following topics:
- Relational Database Basics
- Attributes and Tuples
- Keys in Relational Databases
- Definition and Properties of Functional Dependencies
- Armstrong’s Axioms
- Normalization and its Forms (1NF, 2NF, 3NF, BCNF)
Here’s a simple example to illustrate functional dependencies:
| StudentID | Name | Age |
|---|---|---|
| 1 | Alice | 20 |
| 2 | Bob | 22 |
| 3 | Charlie | 21 |
In this table:
- StudentID → Name (StudentID uniquely determines Name)
- StudentID → Age (StudentID uniquely determines Age)
- Name ↛ Age (Name does not uniquely determine Age, as different students could have the same age)
