CS Free MCQ Bank

[ Option C ]

Given the relation R(A, B, C, D) with functional dependencies:
A → B
B → C
C → D
Primary Key: A
Check for BCNF:
A relation is in Boyce-Codd Normal Form (BCNF) if for every functional dependency X→Y,  X is a candidate key or super key.
So, BCNF ensures that each non—key (non—prime) attributes is dependent only on the candidate key.
A→B: A is the primary key (Super Key): Satisfy BCNF condition.
B→C: B is not a Super Key : Violates BCNF condition.
C→D: C is not a Super Key : Violates BCNF condition.
Check for 3NF:
The relation is in 3NF if there is no transitive dependency i.e., no non—prime attribute can determine another non—prime attribute.
A→B: A is the prime attribute : Satisfy 3NF condition.
B→C: Both B and C are non—prime attributes : Violates 3NF condition.
C→D: Both C and D are non—prime attributes : Violates 3NF condition.
Check for 2NF:
The relation is in 2NF if there is no partial dependency i.e., no proper subset of candidate key can determine the non—prime attribute. 
The relation is at least in 2NF, if key contains only one attribute (here A is primary key, only one attribute no subset possible), then there is no partial dependency because one attribute has no subset. So, the relation is in 2NF. Among the given options the correct answer is The relation is not in BCNF.

[ Option B ]

Normalization up to Third Normal Form (3NF) is carried out in three logical steps.
First, the table is converted to First Normal Form (1NF) by removing repeating groups and multivalued attributes so that each field contains atomic values.
Next, the table is converted to Second Normal Form (2NF) by removing partial dependencies, ensuring that every non-key attribute is fully dependent on the entire primary key.
Finally, the table is converted to Third Normal Form (3NF) by removing transitive dependencies, so that no non-prime attribute can determine another non-prime attribute.

[ Option A ]

A Functional Dependency (FD) describes a relationship where one set of attributes uniquely determines another set of attributes in a relation. 
Key Dependency is a special case of functional dependency in which a key attribute determines all other attributes in the table.

[ Option C ]

To see which dependencies follow from the given ones, compute attribute closures using the given FDs: P→QR and RS→T. If an attribute set’s closure contains the right-hand side, that dependency is implied.
(a) 

P→R
Find P⁺ :
P⁺ = {P}
P⁺={P,Q,R} Using P→QR
R is in P+ ⇒ P→R inferred.
(b)

PS → T
Find PS⁺ :
(PS)⁺={P,S}
(PS)⁺={P,S,Q,R} Using P → QR
Now, (PS)⁺ has RS so Using RS→T
(PS)⁺={P,S,Q,R,T}
T is in (PS)⁺ ⇒ PS → T inferred.
(c)

R→T
Find R⁺ : 
R⁺ = {R}
No dependency with just R, so cannot go further.
T is not in R⁺ ⇒ R→T cannot be inferred.
(d)

PS → Q
Find PS⁺ : 
(PS)⁺ = {P,S)
(PS)⁺ = {P,S,Q,R) Using P→QR
Now, (PS)⁺ has RS so Using RS→T
(PS)⁺={P,S,Q,R,T}
Q is in (PS)⁺ ⇒ PS → Q inferred.

[ Option C ]

A relation is said to be in First Normal Form (1NF) when each attribute contains only atomic or indivisible values. This means there are no repeating groups or multivalued attributes in the table.

[ Option B ]

In the context of database design, Normalization is a process used to organize data efficiently within a database, reducing redundancy and improving data integrity.

For the given relation R(A,B,C,D,E,F)  with functional dependencies AB→C, C→ABDE, and ADE→F. We first identify the candidate keys. Here, C and AB are candidate keys because each can uniquely identify all attributes.
The prime attributes (part of key) are A, B, and C, while D, E, and F are non-prime.

To check which normal form the relation satisfies, we start with 2NF (Second Normal Form), which requires that all non-key (non-prime) attributes are fully functionally dependent on the entire primary key. Since no partial dependency exists here, the relation is in 2NF.
For 3NF, every functional dependency X→Y must satisfy means X is a superkey or Y is a prime attribute.

However, the dependency ADE→F violates this because ADE is not a superkey, and  F is not a prime attribute. This means the relation does not satisfy 3NF.
Finally, since the relation is not in 3NF, it also is not in BCNF (Boyce-Codd Normal Form), which is stricter than 3NF.

[ Option B ]

Armstrong’s Axioms are a set of inference rules used to derive all functional dependencies (FDs) logically implied by a given set of FDs in a database.

Statement I says that if a functional dependency B → C holds, then adding the same set of attributes A to both sides results in AB → AC. This is known as the Augmentation Rule. Statement II says that if B ⊆ A (B is a subset of A), then A → B always holds, which is called the Reflexivity Rule.

[ Option A ]

The decomposition of schema R(A,B,C,D) with functional dependencies A→B and C→D into R1(AB) and R2(CD), is dependency preserving but not lossless join.
Lossless Join: A decomposition is lossless if you can always recover the original relation by joining the decomposed relations, without losing any data. For a lossless join, the intersection of the schemas must be a key in at least one of the decomposed relations.
Dependency Preserving: A decomposition is dependency preserving if every functional dependency in the original relation is still enforceable in at least one of the decomposed relations without needing to join.

• The intersection between R1 and R2 is empty (AB∩CD=∅), so the lossless join condition fails.
• The dependencies A→B and C→D can still be enforced within R1 and R2 respectively. So, these dependencies are preserved.

[ Option B ]

A relation is in Third Normal Form (3NF) if:

• It is in Second Normal Form (2NF).
• Every non-key attribute is fully functionally dependent on the primary key.
• It has no transitive dependencies — meaning no non-key attribute depends on another non-key attribute.

This means each non-key attribute depends only on the primary key and not on other non-key attributes, which helps eliminate redundancy and update anomalies.

[ Option B ]

In DBMS, a Functional Dependency (FD) represents a relationship between two sets of attributes in a relation. It is written as X → Y, meaning that the value of attribute set X uniquely determines the value of attribute set Y.

A Trivial functional dependency is one that always holds true and does not provide any new information about the data. It occurs when the dependent attributes (Y) are already included in the determinant (X). In other words, if Y is a subset of X (Y ⊆ X), the dependency is called trivial.

• For example, dependencies like A → A, AB → A and ABC → AB are trivial because the right side is included within the left side.

[ Option D ]

DKNF (Domain-Key Normal Form), is an advanced level of database normalization. A table is said to be in DKNF when all rules about the data can be enforced using only domain rules and key rules. 
Domain rules decide what type of values are allowed in a column, and key rules decide how rows are uniquely identified.

[ Option C ]

In database normalization, BCNF stands for Boyce–Codd Normal Form. BCNF was proposed by Raymond F. Boyce and Edgar F. Codd.
A relation is said to be in BCNF if for every functional dependency, the determinant is a super key.
Means, a relation is in BCNF if for every functional dependency X→Y,  X is a candidate key or super key.
So, BCNF ensures that each non—key (non—prime) attributes is dependent only on the candidate key.

[ Option B ]

To determine the highest normal form of the relation S(S1,S2,S3,E1,E2) with the given functional dependencies:
FD1: S1→{S2,S3}
FD2: {E1,S3,S2}→S1
FD3: E2→{S3,S2,E1}
First of all, find the key. Using the above functional dependencies the E2 determine all the attributes, so, E2 is candidate key. That makes E2 the only prime attribute.
Prime Attribute : E2.
Non-Prime Attribute : S1, S2, S3, E1.

Check 2NF:
The relation is at least in 2NF, if key contains only one attribute, then there is no partial dependency because one attribute has no subset. So, the relation is in 2NF.

• If the proper subset of candidate key determines non-prime attribute, it is called Partial Dependency.

Check 3NF:
The relation is called in 3NF it is already exists in 2NF and there is no transitive dependency exists.

• If non-prime attribute can determine another non-prime attribute, then it is called Transitive Dependency.

In the given relation FD set, the S1 can determine S2 and S3. The S1, S2 and S3 are non-prime attributes so these violets the rule of transitive dependency.
So, the highest normal form the relation satisfies is Second Normal Form (2NF).

[ Option A ]

A table is said to be in Second Normal Form (2NF) when it first satisfies the conditions of First Normal Form (1NF) and, in addition, every non-key attribute is fully functionally dependent on the entire primary key.

[ Option C ]

A Multivalued Dependency (MVD) occurs when one attribute in a table determines multiple independent sets of values for another attribute. Fourth Normal Form (4NF) is specifically designed to eliminate multivalued dependencies from a relation.
Consider the table,

Here:

• A student can enroll in multiple courses.
• The same student can have multiple hobbies.
• Courses and hobbies are independent of each other.

This creates a multivalued dependency:
Rollno→Course
Rollno→Hobby
Because of this, data is repeated unnecessarily.

• 1NF removes repeating groups and ensures Atomic Values.
• 2NF removes Partial Dependency.
• 3NF removes Transitive Dependency.

[ Option B ]

In First Normal Form (1NF), a table must be organized in such a way that there are no repeating groups or multivalued attributes. This means that each field should contain only atomic values, and each column should store values of a single type.

[ Option D ]

Armstrong’s Axioms are a set of inference rules used to derive all functional dependencies (FDs) logically implied by a given set of FDs in a database.
The three fundamental axioms are:

1. Reflexivity Rule.
2. Augmentation Rule.
3. Transitivity Rule.

The Union Rule is not a basic axiom. It is a derived rule that can be obtained using Armstrong’s axioms.

[ Option B ]

Normalization is a database design technique introduced by Dr. E. F. Codd, the founder of the relational model. Normalization helps in reducing data redundancy and eliminating anomalies by organizing data into well-structured tables.

[ Option A ]

A Minimal Cover or Canonical Cover for a set of functional dependencies is an equivalent set where every dependency has a single attribute on the right-hand side, every left-hand side has no extraneous attribute, and no dependency is redundant.

[ Option A ]

BCNF is an advanced version of 3NF used to remove redundancy caused by functional dependencies. A relation R is in BCNF with respect to a set of functional dependencies F if for every non-trivial functional dependency A → B in F+, the determinant A is a superkey of R.