Given a relation scheme R(x, y, z, w) with functional dependencies set F = {x → y, z → w}. All attributes take single and atomic values only. A. Relation R is in First Normal FORM B. Relation R is in Second Normal FORM C. Primary key of R is xz Choose the correct answer from the options given below :
A and C only
Understanding Relation R and Functional Dependencies
The problem describes a relational scheme R with attributes x, y, z, and w. We are given a set of functional dependencies (FDs): {x → y, z → w}. It's also stated that all attributes take single and atomic values. We need to evaluate three statements about this relation: its normal form and its primary key.
Analyzing Statement A: Relation R is in First Normal Form (1NF)
A relation is in First Normal Form (1NF) if and only if all attribute values are atomic and single-valued. The problem explicitly states that "All attributes take single and atomic values only." This directly matches the definition of 1NF.
Therefore, Statement A is true.
Determining the Primary Key of Relation R
A primary key is a minimal set of attributes that uniquely identifies each tuple in the relation. We can find candidate keys by examining the functional dependencies.
- Attributes on the right-hand side (RHS) of the FDs are {y, w}. These attributes are determined by other attributes.
- Attributes not on the RHS of any FD are {x, z}. These attributes must be part of any candidate key, as they cannot be determined by other attributes.
Let's check the closure of {x, z} to see if it can determine all attributes in R:
{x, z}<sup>+</sup>
- Initially, {x, z}<sup>+</sup> = {x, z}.
- From x → y, since x is in {x, z}<sup>+</sup>, we add y. {x, z}<sup>+</sup> = {x, z, y}.
- From z → w, since z is in {x, z}<sup>+</sup>, we add w. {x, z}<sup>+</sup> = {x, z, y, w}.
Since {x, z}<sup>+</sup> = {x, y, z, w}, the set {x, z} is a candidate key. To be a primary key, it must be minimal. Can we remove x or z and still determine R?
- {x}<sup>+</sup> = {x, y}. Does not contain z or w. {x} is not a key.
- {z}<sup>+</sup> = {z, w}. Does not contain x or y. {z} is not a key.
Since neither {x} nor {z} is a key, {x, z} is a minimal candidate key. Therefore, {x, z} can be the primary key of R.
Thus, Statement C is true.
Analyzing Statement B: Relation R is in Second Normal Form (2NF)
A relation is in Second Normal Form (2NF) if it is in 1NF and every non-prime attribute is fully functionally dependent on every candidate key.
- We've established R is in 1NF.
- The candidate key is {x, z}.
- Prime attributes are the attributes that are part of the candidate key: x, z.
- Non-prime attributes are the attributes not part of the candidate key: y, w.
Now we check if the non-prime attributes (y, w) are fully dependent on the candidate key {x, z}. This means we must check for partial dependencies, where a non-prime attribute is dependent on a proper subset of a candidate key.
- Consider the non-prime attribute y. We have the FD x → y. The determinant x is a proper subset of the candidate key {x, z}. This is a partial dependency.
- Consider the non-prime attribute w. We have the FD z → w. The determinant z is a proper subset of the candidate key {x, z}. This is another partial dependency.
Since there are partial dependencies (x → y and z → w) where non-prime attributes are dependent on a proper subset of the candidate key, the relation R is not in 2NF.
Therefore, Statement B is false.
Summary of Findings
Based on our analysis:
- Statement A: Relation R is in First Normal FORM - True
- Statement B: Relation R is in Second Normal FORM - False
- Statement C: Primary key of R is xz - True
The correct answer corresponds to statements A and C being true, and B being false.
| Statement | Evaluation | Reason |
|---|---|---|
| A. Relation R is in First Normal FORM | True | Attributes are single and atomic valued (given). |
| B. Relation R is in Second Normal FORM | False | Existence of partial dependencies (x → y, z → w) on the candidate key {x, z}. |
| C. Primary key of R is xz | True | {x, z} is a minimal set of attributes that determines all other attributes in R. |
Revision Table: Database Normalization Concepts
| Normal Form | Requirements | Explanation |
|---|---|---|
| 1NF (First Normal Form) | All attribute values must be atomic and single-valued. No repeating groups. | Ensures data is stored in a basic table structure with no complex data types or multi-valued attributes within a single cell. |
| 2NF (Second Normal Form) | Must be in 1NF and every non-prime attribute must be fully functionally dependent on every candidate key. | Eliminates partial dependencies by ensuring non-key attributes depend on the entire candidate key, not just a part of it. |
| 3NF (Third Normal Form) | Must be in 2NF and no non-prime attribute is transitively dependent on a candidate key. | Eliminates transitive dependencies by ensuring non-key attributes do not depend on other non-key attributes. |
Additional Information: Functional Dependencies and Keys
Functional dependencies (FDs) are constraints between attributes in a relation that specify how one attribute or set of attributes determines another. They are fundamental to database design and normalization.
- Determinant: The attribute(s) on the left-hand side of an FD (e.g., x in x → y).
- Dependent: The attribute(s) on the right-hand side of an FD (e.g., y in x → y).
- Candidate Key: A set of attributes K is a candidate key for a relation R if:
- K functionally determines all attributes in R (K → R).
- K is minimal (no proper subset of K functionally determines R).
- Primary Key: One of the candidate keys chosen by the database designer to uniquely identify tuples.
- Prime Attribute: An attribute that is part of *any* candidate key.
- Non-prime Attribute: An attribute that is not part of *any* candidate key.
Understanding these concepts is crucial for evaluating the normal form of a relation and ensuring a good database design that minimizes redundancy and anomalies.
Important Questions from Normal Form - Teaching
Consider a schema R(A, B, C, D) and following functional dependencies.
A → B
B → C
C → D
D → B
Then decomposition of R into R 1(A, B), R 2(B, C) and R 3(B, D) is _______
If every non-key attribute is functionally dependent on the primary key, then the relation is in _______.
Consider a relation R (A, B, C, D, E, F, G, H), where each attribute is atomic, and following functional dependencies exist.
CH → G
A → BC
B → CFH
E → A
F → EG
The relation R is ______.
For a database relation R(a, b, c, d) where the domains of a, b, c, and d include only atomic values, and only the following functional dependencies and those that can be inferred from them hold:
a → c
b → d
The relation is in.Match List I with List II:
List I List II (A) BCNF (I) It removes multivalued dependency (B) 3 NF (II) It is not always dependency preserving (C) 2 NF (III) It removes transitive dependency (D) 4 NF (IV) It removes partial functional dependency Choose the correct answer from the options given below:
