GATE CS Normal Forms questions

Let $P, Q, R$ and $S$ be the attributes of a relation in a relational schema. Let $X \rightarrow Y$ indicate functional dependency in the context of a relational database, where $X, Y \subseteq\{P, Q, R, S\}$. Which of the following options is/are always true?

In the context of relational database normalization, which of the following statements is/are true?

In the context of schema normalization in relational DBMS, consider a set $\text{F}$ of functional dependencies. The set of all functional dependencies implied by $\text{F}$ is called the closure of $\text{F}$. To compute the closure of $\text{F}$, Armstrong's Axioms can be applied. Consider $X, Y$, and $Z$ as sets of attributes over a relational …

Consider a relational schema team $\text{(name,city,owner)}$, with functional dependencies $\{\text{name} \rightarrow \text{city}, \text{name} \rightarrow \text{owner} \}$. The relation team is decomposed into two relations, $\text{t1(name,city)}$ and $\text{t2(name, owner)}$. Which of the following statement(s) is/are TRUE?

​​​​Consider the following relational schema along with all the functional dependencies that hold on them. $$\begin{array}{l} R 1(A, B, C, D, E):\{D \rightarrow E, E A \rightarrow B, E B \rightarrow C\} \\ R 2(A, B, C, D):\{A \rightarrow D, A \rightarrow B, C \rightarrow A\} \end{array}$$ Which of the following statement(s) is/are TRUE?

Which of the following statements about a relation $\mathbf{R}$ in first normal form $\text{(1NF)}$ is/are TRUE?

​The symbol $\rightarrow$ indicates functional dependency in the context of a relational database. Which of the following options is/are TRUE?

A functional dependency $F: X \rightarrow Y$ is termed as a useful functional dependency if and only if it satisfies all the following three conditions: $\text{X}$ is not the empty set. $\text{Y}$ is not the empty set. Intersection of $\text{X}$ and $\text{Y}$ is the empty set. For a relation $\text{R}$ with $4$ attributes, the total number of …

In a relational data model, which one of the following statements is $\text{TRUE}?$

Consider a relation $R (A, B, C, D, E)$ with the following three functional dependencies. $AB \rightarrow C; \; BC \rightarrow D; \; C \rightarrow E;$ The number of superkeys in the relation $R$ is ______________ .

Consider the relation $R(P,Q,S,T,X,Y,Z,W)$ with the following functional dependencies. $$PQ\rightarrow X;\quad P\rightarrow YX;\quad Q\rightarrow Y; \quad Y\rightarrow ZW$$ Consider the decomposition of the relation $R$ into the constituent relations according to the following two decomposition schemes. …

Suppose the following functional dependencies hold on a relation $U$ with attributes $P,Q,R,S$, and $T$: $P \rightarrow QR$ $RS \rightarrow T$ Which of the following functional dependencies can be inferred from the above functional dependencies?

Consider a relational table $R$ that is in $3NF$, but not in BCNF. Which one of the following statements is TRUE?

Let the set of functional dependencies $F=\{QR \rightarrow S, \: R \rightarrow P, \: S \rightarrow Q \}$ hold on a relation schema $X=(PQRS)$. $X$ is not in BCNF. Suppose $X$ is decomposed into two schemas $Y$ and $Z$, where $Y=(PR)$ and $Z=(QRS)$. Consider the two statements given below. Both $Y$ and $Z$ are in BCNF Decomposition of …

Consider the following four relational schemas. For each schema , all non-trivial functional dependencies are listed, The bolded attributes are the respective primary keys. $\text{Schema I:}$ Registration(rollno, courses) Field ‘courses’ is a set-valued attribute containing the set of courses a student has registered for. Non-trivial functional dependency …

The following functional dependencies hold true for the relational schema $R\left \{V,W,X,Y,Z \right \}$: $V \rightarrow W$ $VW \rightarrow X$ $Y \rightarrow VX$ $Y \rightarrow Z$ Which of the following is irreducible equivalent for this set of functional dependencies?

Which of the following is NOT a superkey in a relational schema with attributes $V,W,X,Y,Z$ and primary key $V\;Y$?

Consider two relations $R_1(A,B)$ with the tuples $(1,5), (3,7)$ and $R_2(A,C) = (1,7),(4,9)$. Assume that $R(A,B,C)$ is the full natural outer join of $R_1$ and $R_2$. Consider the following tuples of the form $(A,B,C):$ $a = (1,5,null), b=(1,null,7), c=(3,null,9), d=(4,7,null), e=(1,5,7),$ $f=(3,7,null), g=(4,null,9).$ Which one of the …

Consider the relation $X(P,Q,R,S,T,U)$ with the following set of functional dependencies $F = \{ \\ \; \; \{P, R\} \rightarrow \{S, T\}, \\ \; \; \{P, S, U\} \rightarrow \{Q, R\} \\ \; \}$ Which of the following is the trivial functional dependency in $F^+$, where $F^+$ is closure to F?

Given the following two statements: S1: Every table with two single-valued attributes is in $\text{1NF, 2NF, 3NF}$ and $\text{BCNF}.$ S2: $AB \to C, D \to E, E \to C$ is a minimal cover for the set of functional dependencies $AB \to C, D \to E, AB \to E, E \to C$. Which one of the following is CORRECT?