GATE CS Relational Model: Relational Algebra questions

Consider the following two relations, $R(A, B)$ and $S(A, C)$: …

​​​​​The relation schema, Person (pid, city), describes the city of residence for every person uniquely identified by pid. The following relational algebra operators are available: selection, projection, cross product, and rename. To find the list of cities where at least $3$ persons reside, using the above operators, the minimum number of cross product …

Which one of the options given below refers to the degree (or arity) of a relation in relational database systems?

Consider the following three relations in a relational database. $\text{Employee} (\underline{\text{eId}},\text{Name}), \; \text{Brand}(\underline{\text{bId}},\text{bName}), \; \text{Own}(\underline{\text{eId}},\underline{\text{bId}})$ Which of the following relational algebra expressions return the set of $\textit{elds}$ who own all the brands?

The following relation records the age of $500$ employees of a company, where $empNo$ (indicating the employee number) is the key: $$empAge(\underline{empNo},age)$$ Consider the following relational algebra expression: $$\Pi_{empNo}(empAge \Join_{(age>age1)} \rho_{empNo1,age1}(empAge))$$ What does the above expression generate?

Consider the following relations $P(X,Y,Z), Q(X,Y,T)$ and $R(Y,V)$. …

Consider the relations $r(A, B)$ and $s(B, C)$, where $s.B$ is a primary key and $r.B$ is a foreign key referencing $s.B$. Consider the query $Q: r \bowtie (\sigma_{B<5} (s))$ Let LOJ denote the natural left outer-join operation. Assume that $r$ and $s$ contain no null values. Which of the following is NOT equivalent to $Q$?

Consider a database that has the relation schema CR(StudentName, CourseName). An instance of the schema CR is as given below. …

What is the optimized version of the relation algebra expression $\pi_{A1}(\pi_{A2}(\sigma_{F1}(\sigma_{F2}(r))))$, where $A1, A2$ are sets of attributes in $r$ with $A1 \subset A2$ and $F1,F2$ are Boolean expressions based on the attributes in $r$?

Consider the following relations $A, B$ and $C:$ …

Suppose $R_{1} (\underline{A}, B)$ and $R_{2} (\underline{C}, D)$ are two relation schemas. Let $r_{1}$ and $r_{2}$ be the corresponding relation instances. $B$ is a foreign key that refers to $C$ in $R_{2}$. If data in $r_{1}$ and $r_{2}$ satisfy referential integrity constraints, which of the following is ALWAYS TRUE?

Let R and S be two relations with the following schema $R(\underline{P,Q}, R1, R2, R3)$ $S(\underline{P,Q}, S1, S2)$ where $\left\{P, Q\right\}$ is the key for both schemas. Which of the following queries are equivalent? $\Pi_P \left(R \bowtie S\right)$ $\Pi_P \left(R\right) \bowtie \Pi_P\left(S\right)$ …

Information about a collection of students is given by the relation $\text{studInfo(}\underline{\text{studId}},\text{ name, sex)}$. The relation $\text{enroll(}{\text{studId}},{\text{ courseId}})$ gives which student has enrolled for (or taken) what course(s). Assume that every course is taken by at least one male and at least one female student. What …

Let r be a relation instance with schema R = (A, B, C, D). We define $r_1 = \pi_{A, B, C} (R)$ and $r_2=\pi_{A, D} (r)$. Let $s =r_1 \: * \: r_2$ where $*$ denotes natural join. Given that the decomposition of $r$ into $r_1$ and $r_2$ is lossy, which one of the following is TRUE?

Consider the following relation schema pertaining to a students database: Students (rollno, name, address) Enroll (rollno, courseno, coursename) where the primary keys are shown underlined. The number of tuples in the student and Enroll tables are $120$ and $8$ respectively. What are the maximum and minimum number of tuples that can be present in …

Consider the following SQL query Select distinct $a_1, a_2, …, a_n$ from $r_1, r_2, …, r_m$ where P For an arbitrary predicate P, this query is equivalent to which of the following relational algebra expressions?

Let r and s be two relations over the relation schemes R and S respectively, and let A be an attribute in R. The relational algebra expression $\sigma_{A=a}(r \bowtie s)$ is always equal to

Suppose the adjacency relation of vertices in a graph is represented in a table Adj $(X,Y).$ Which of the following queries cannot be expressed by a relational algebra expression of constant length?

Given the relations employee (name, salary, dept-no), and department (dept-no, dept-name,address), Which of the following queries cannot be expressed using the basic relational algebra operations $\left(\sigma, \pi,\times ,\Join, \cup, \cap,-\right)$?

Consider the join of a relation $R$ with a relation $S$. If $R$ has $m$ tuples and $S$ has $n$ tuples then the maximum and minimum sizes of the join respectively are