RelationalAlgebra
Question 1 
Consider the following relations P(X,Y,Z), Q(X,Y,T) and R(Y,V).
How many tuples will be returned by the following relational algebra query?
∏x(σ(P.Y=R.Y ∧ R.V=V2)(P × R))  ∏x(σ(Q.Y=R.Y ∧ Q.T>2)(Q × R))
0  
1  
2  
3 
∏_{x}(σ_{(P.Y = R.Y ∧ R.V = V2)}(P × R))
σ_{(Q.Y = R.Y ∧ Q.T>2)}(Q × R)
∏_{x}(σ_{(Q.Y = R.Y ∧ Q.T>2)}(Q × R))
∏_{x}(σ_{(P.Y = R.Y ∧ R.V = V2)}(P × R))  ∏_{x}(σ_{(Q.Y = R.Y ∧ Q.T>2)}(Q × R))
Question 2 
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⋈(σ_{B<5}(s))
Let LOJ denote the natural left outerjoin operation. Assume that r and s contain no null values.
Which one of the following is NOT equivalent to Q?
σ_{B<5} (r ⨝ s)  
σ_{B<5} (r LOJ s)
 
r LOJ (σ_{B<5}(s))
 
σ_{B<5}(r) LOJ s 
Consider the following tables without NULL values.
Q: r⨝(σ_{B}<5(S))
The result of σ_{B<5}(S) is
The result of σ_{B<5}(S) is
Option (A):
The result of r⨝S is
The result of σ_{B<5}(r⨝S) is
Option (B):
The result of r LOJ S is
The result of σ_{B<5}(r LOJ S) is
Option (C):
The result of σ_{B<5}(S) is
Now, the result of r LOJ(σ_{B<5}(S))
Option (D):
The result of σ_{B<5}(r) is
Now, the result of σ_{B<5}(r) LOJ S is
Therefore, from the output of above four options, the results of options, the results of options (A), (B) and (D) are equivalent to Q.
Question 3 
Consider a database that has the relation schemas EMP(EmpId, EmpName, DeptId), and DEPT(DeptName, DeptId). Note that the DeptId can be permitted to a NULL in the relation EMP. Consider the following queries on the database expressed in tuple relational calculus.

(I) {t│∃u ∈ EMP(t[EmpName] = u[EmpName] ∧ ∀v ∈ DEPT(t[DeptId] ≠ v[DeptId]))}
(II) {t│∃u ∈ EMP(t[EmpName] = u[EmpName] ∧ ∃v ∈ DEPT(t[DeptId] ≠ v[DeptId]))}
(III) {t│∃u ∈ EMP(t[EmpName] = u[EmpName] ∧ ∃v ∈ DEPT(t[DeptId] = v[DeptId]))}
Which of the above queries are safe?
(I) and (II) only  
(I) and (III) only  
(II) and (III) only  
(I), (II) and (III) 
(I) Gives EmpNames who do not belong to any Department. So, it is going to be a finite number of tuples as a result.
(II) Gives EmpNames who do not belong to some Department. This is also going to have finite number of tuples.
(III) Gives EmpNames who do not belong to same Department. This one will also give finite number of tuples.
All the expressions I, II and III are giving finite number of tuples. So, all are safe.
Question 4 
Consider a database that has the relation schema CR(StudentName, CourseName). An instance of the schema CR is as given below.
The following query is made on the database.
T1 ← π_{CourseName}(σ_{StudentName='SA'}(CR)) T2 ← CR ÷ T1
The number of rows in T2 is ____________.
4  
5  
6  
7 
The σ_{StudentName = 'SA'}(CR) will produce the following
⇾ The result of T1 ← π_{CourseName}(σ_{StudentName='SA'}(CR)) is
(2) T2 ← CR÷T1
⇾ We see that SA is enrolled for CA, CB and CC.
⇾ T2 will give the StudentNames those who have enrolled for all the CA, C, CC courses. So, the following Students are enrolled for the given 3 courses.
⇾ So, the output of T2 will have 4 rows.
Question 5 
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 following statements is correct?
R contains a,b,e,f,g but not c, d.
 
R contains all of a,b,c,d,e,f,g  
R contains e,f,g but not a,b  
R contains e but not f,g 
⋆ So, from the above resultant table, R contains e, f, g only but not a, b.
Question 6 
Suppose R_{1}(A, B) and R_{2}(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?
∏_{B} (r_{1})  ∏_{C} (r_{2}) = ∅  
∏_{C} (r_{2})  ∏_{B} (r_{1}) = ∅  
∏_{B} (r_{1}) = ∏_{C} (r_{2})  
∏_{B} (r_{1})  ∏_{C} (r_{2}) ≠ ∅ 
So we can say that r_{2}(C) is the superset of r_{1}(B).
So (subset  superset) is always empty.
Question 7 
Consider the following relations A, B, C.
How many tuples does the result of the following relational algebra expression contain? Assume that the schema of AUB is the same as that of A.
(AUB)⋈_{A.Id>40∨C.Id<15} C
7  
4  
5  
9 
Performs the cross product and selects the tuples whose A∙Id is either greater than 40 or C∙Id is less than 15. It yields:
Question 8 
Which of the following functional dependencies hold for relations R(A, B, C) and S(B, D, E):
B > A A > C
The relation R contains 200 tuples and the rel ation S contains 100 tuples. What is the maximum number of tuples possible in the natural join of R and S (R natural join S)
100  
200  
300  
2000 
R(A, B, C) – 200 tuples
S(B, D, E) – 100 tuples
FD’s:
B → A
A → C
― ‘B’ is primary key for R and foreign key of S from the given FDs.
― Maximum tuples in natural join of R and S is min(200, 100) = 100.
Question 9 
Let R and S be two relations with the following schema
 R(P,Q,R1,R2,R3)
S(P,Q,S1,S2)
Where {P, Q} is the key for both schemas. Which of the following queries are equivalent?
 I. Π_{P}(R ⨝ S)
II. Π_{P}(R) ⨝ Π_{P}(S)
III. Π_{P}(Π_{P,Q}(R) ∩ Π_{P,Q}(S))
IV. Π_{P}(Π_{P,Q}(R)  (Π_{P,Q}(R)  Π_{P,Q}(S)))
Only I and II  
Only I and III  
Only I, II and III  
Only I, III and IV 
We have two common columns in 'R' and 'S' which are 'P' and 'Q'.
(I) Both P and Q are used while doing the join, i.e., both P and Q are used to filter.
(II) Q is not used here for filtering. Natural join is done on all P's from R and all P's from S. So different from option (I).
(III) Through venn diagram it can be proved that A∩B = A  (AB).
So through above formula we can say that (III) and (IV) are equivalent.
So, finally (I), (III) and (IV) are equivalent.
Question 10 
Information about a collection of students is given by the relation studinfo(studId, name, sex). The relation enroll(studId, courseId) gives which student has enrolled for (or taken) that course(s). Assume that every course is taken by at least one male and at least one female student. What does the following relational algebra expression represent?
Π_{courseId}((Π_{studId}(σ_{sex="female"}(studInfo))×Π_{courseId}(enroll))enroll)Courses in which all the female students are enrolled.  
Courses in which a proper subset of female students are enrolled.  
Courses in which only male students are enrolled.
 
None of the above 
Option B: Yes, True. It selects the proper subset of female students which are enrolled because in the expression we are performing the Cartesian product.
Option C: False. It doesn’t shows (or) display the males students who are enrolled.
Question 11 
Let r be a relation instance with schema R = (A, B, C, D). We define r_{1} = Π_{A,B,C}(R) and r_{2} = Π_{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?
s ⊂ r  
r ∪ s = r  
r ⊂ s  
r * s = s

Table r: R(A, B, C, D)
Table r_{1}: Π_{A,B,C}(R)
Table r_{2}: Π_{A,D}(R)
S = r_{1} * r_{2} (* denotes natural join)
Table S:
Table r ⊂ Table S
⇒ r ⊂ S
Question 12 
Let R_{1}(A,B,C) and R_{2}(D,E) be two relation schema, where the primary keys are shown underlined, and let C be a foreign key in R_{1} referring to R_{2}. Suppose there is no violation of the above referential integrity constraint in the corresponding relation instances r_{1} and r_{2}. Which one of the following relational algebra expressions would necessarily produce an empty relation?
Π_{D}(r_{2})  Π_{C}(r_{1})  
Π_{C}(r_{1})  Π_{D}(r_{2})  
Π_{D}(r_{1}⨝_{C≠D}r_{2})  
Π_{C}(r_{1}⨝_{C=D}r_{2}) 
→ Based on referral integrity C is subset of values in R_{2} then,
Π_{C}(r_{1})  Π_{D}(r_{2}) results empty relation.
Question 13 
Consider the following relation schema pertaining to a students database:
Student (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 (Student * Enroll), where '*' denotes natural join ?
8, 8  
120, 8  
960, 8  
960, 120 
→ In the question only enroll Id's are same with the student table.
→ The no. of minimum and maximum tuples is same i.e., 8, 8.
Question 14 
Consider the relation Student (name, sex, marks), where the primary key is shown underlined, pertaining to students in a class that has at least one boy and one girl. What does the following relational algebra expression produce? (Note: p is the rename operator).
The condition in join is "(sex = female ^ x = male ^ marks ≤ m)"
names of girl students with the highest marks
 
names of girl students with more marks than some boy student  
names of girl students with marks not less than some boy students
 
names of girl students with more marks than all the boy students 
Question 15 
With regard to the expressive power of the formal relational query languages, which of the following statements is true?
Relational algebra is more powerful than relational calculus.  
Relational algebra has the same power as relational calculus.  
Relational algebra has the same power as safe relational calculus.  
None of the above. 
A query can be formulated in safe Relational Calculus if and only if it can be formulated in Relational Algebra.
Question 16 
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?
List of all vertices adjacent to a given vertex  
List all vertices which have self loops  
List all vertices which belong to cycles of less than three vertices  
List all vertices reachable from a given vertex 
(b) Finding a self loop is also simple (Oop(X,X))
(c) If a → b, b → c then c!=a, finding this is also simple.
(d) List all the elements reachable from a given vertex is too difficult in Relational Algebra.
Question 17 
Let r and s be two relations over the relation schemes R and S respectively, and let A be an attribute in R. Then the relational algebra expression σ_{A=a}(r⋈s) is always equal to
σ_{A=a} (r)  
r  
σ_{A=a} (r)⨝s  
None of the above 
(b) Display table
(c) A=a for all Tables r and s
Question 18 
Given the relations
employee (name, salary, deptno) and department (deptno, deptname, address)
Which of the following queries cannot be expressed using the basic relational algebra operations (σ, π, ×, ⋈, ∪, ∩, )?
Department address of every employee  
Employees whose name is the same as their department name  
The sum of all employees’ salaries  
All employees of a given department 
Question 19 
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
m + n and 0  
mn and 0  
m + n and m – n  
mn and m + n 
Suppose there is no common attribute in R and S due to which natural join will act as cross product. So then in cross product total no. of tuples will be mn.
For minimum:
Suppose there is common attribute in R and S, but none of the row of R matches with rows of S then minimum no. of tuples will be 0.
Question 20 
The relational algebra expression equivalent to the following tuple calculus expression:
{t t ∈ r ∧(t[A] = 10 ∧ t[B] = 20)} isσ_{(A=10∨B=20)} (r)  
σ_{(A=10)} (r) ∪ σ_{(B=20)} (r)  
σ_{(A=10)} (r) ∩ σ_{(B=20)} (r)  
σ_{(A=10)} (r)  σ_{(B=20)} (r) 
σ_{(A=10)} (r) ∩ σ_{(B=20)} (r)
Question 21 
Given two union compatible relations R_{1}(A,B) and R_{2}(C,D), what is the result of the operation R_{1}A = CAB = DR_{2}?
R_{1} ∪ R_{2}  
R_{1} × R_{2}  
R_{1}  R_{2}  
R_{1} ∩ R_{2} 
Question 22 
Which of the following query transformations (i.e. replacing the l.h.s. expression by the r.h.s. expression) is incorrect? R_{1} and R_{2} are relations, C_{1}, C_{2} are selection conditions and A_{1}, A_{2} are attributes of R_{1}?
σ_{C1}(σ_{C1}(R_{1})) → σ_{C2}(σ_{C2}(R_{1}))  
σ_{C1}(σ_{A1}(R_{1})) → σ_{A1}(σ_{C1}(R_{1}))  
σ_{C1}(R_{1} ∪ R_{2}) → σ_{C1}(R_{1}) ∪ σ_{C1}  
π_{A1}(σ_{C1}(R_{1})) → σ_{C1}(σ_{A1}(R_{1})) 
Question 23 
Give a relational algebra expression using only the minimum number of operators from (∪, −) which is equivalent to R ∩ S.
Out of syllabus (For explanation see below) 
→ No need of using Union operation here. → In question they gave (∪, −) but we don't use both.
→ And also they are saying that only the minimum number of operators from (∪, −) which is equivalent to R ∩ S.
So, the expression is minimal.
Question 24 
Consider a selection of the form σ_{A≤100}(r), where r is a relation with 1000 tuples. Assume that the attribute values for A among the tuples are uniformly distributed in the interval [0, 500]. Which one of the following options is the best estimate of the number of tuples returned by the given selection query ?
50  
100  
150  
200 
Values for A among the tuples are uniformly distributed in the interval [0, 500]. This can be split to 5 mutually exclusive and exhaustive intervals of same width of 100 ([0100], [101200], [201300], [301400], [401500], 0 makes the first interval larger  this must be type in this question) and we can assume all of them have same number of values due to uniform distribution. So no. of tuples with A value in first interval should be,
Total no. of tuples/5 = 1000/5 = 200
Question 25 
Consider the following relation schemas:
bSchema = (bname, bcity, assets)
aSchema = (anum, bname, bal)
dSchema = (cname, anumber)
Let branch, account and depositor be respectively instances of the above schemas. Assume that account and depositor relations are much bigger than the branch relation.
Consider the following query:
П_{cname} (σ_{bcity = "Agra" ⋀ bal < 0} (branch ⋈ (account ⋈ depositor)
Which one of the following queries is the most efficient version of the above query?
П_{cname} (σ_{bal < 0} (σ_{bcity = “Agra”} branch ⋈ account) ⋈ depositor)  
П_{cname} (σ_{bcity = “Agra”}branch ⋈ (σ_{bal < 0} account ⋈ depositor))  
П_{cname} (σ_{bcity = “Agra”} branch ⋈ σ_{bcity = “Agra” ⋀ bal < 0} account) ⋈ depositor)  
П_{cname} (σ_{bcity = “Agra” ⋀ bal < 0} account ⋈ depositor)) 
Options (C) and (D) are invalid as there is no bcity column in aschema.
Question 26 
Consider the relations r_{1}(P, Q, R) and r_{2}(R, S, T) with primary keys P and R respectively. The relation r_{1} contains 2000 tuples and r_{2} contains 2500 tuples. The maximum size of the join r_{1}⋈ r_{2} is :
2000  
2500  
4500  
5000 
Question 27 
Project  
Join  
Extract  
Substitute 
Projection is used to project required column data from a relation. By Default projection removes duplicate data.
Example :
R(A B C)

1 2 4
2 2 3
3 2 3
4 3 4
π (BC)
B C

2 4
2 3
3 4
Question 28 
a cartesian product of two relations followed by a selection  
a cartesian product of two relations  
a union of two relations followed by cartesian product of the two relations  
a union of two relations 
→ The join operation can be defined as a cartesian product of two relations followed by a selection.
→ A SQL JOIN clause is used to combine rows from two or more tables, based on a related column between them.
Different Types of SQL JOINs
1. INNER JOIN: Returns records that have matching values in both tables
2. LEFT (OUTER) JOIN: Return all records from the left table, and the matched records from the right table
3. RIGHT (OUTER) JOIN: Return all records from the right table, and the matched records from the left table
4. FULL (OUTER) JOIN: Return all records when there is a match in either left or right table
Question 29 
m + n & 0  
mn & 0  
m + n &  m – n   
mn & m + n 
If there is common attribute in R and S, and every row of R match with every row of S then total no. of tuples will be mn.
For minimum:
If there is no common attribute between R and S or if there is common attribute but none of the row of R matches with rows of S then output tuples will be 0.
Question 30 
Department address of every employee  
Employees whose name is the same as their department name  
The sum of all employees’ salaries  
All employees of a given department 
Question 31 
Set intersection  
Assignment  
Natural Join  
None of the above 
1.Select
2.Project
3.Cartesian Product
4.Rename
5.Union
6.Set Difference
Question 32 
Set intersection  
Natural Join  
Assignment  
None of the above 
2.Project
3.Cartesian Product
4.Rename
5.Union
6.Set Difference
Question 33 
Consider the schema R = (A, B, C, D, E, F) on which the following functional dependencies hold :
A ➝ B B, C ➝ D E ➝ C D ➝ A
What are the candidate keys of R ?
AEF, BEF and DEF
 
AEF, BEF and BCF  
AE and BE  
AE, BE and DE 
EFB^{+} = {EFABCD}
EFC^{+} = {EFC}
EFD^{+} = {EFDCAB}
So, EFA, EFB, EFD are the keys for the given relation R = (A, B, C, D, E, F).
Question 34 
Record  
Field  
File  
Database 
Equivalent Database Concepts
Relation → Table
Tuple → Row or record
Attribute → Column or field
Cardinality → Number of rows
Degree → Number of columns
Primary key → Unique identifier
Domain → Pool of legal values
Question 35 
R<=dom(A1)X dom(A2)..dom(An)  
R>=dom(A1)X dom(A2)..dom(An)  
R=max(dom(A1),dom(A2),..dom(An))  
R=min(dom(A1),dom(A2),..dom(An)) 
● Cardinality refers to a number. It gives the number of unique values that appear in the table for a particular column.
● For eg: you have a table called Person with column Gender. Gender column can have values either 'Male' or 'Female''.
● Then the cardinality of Gender column is 2, since there are only two unique values that could possibly appear in that column – Male and Female.
Question 36 
join  
self join  
outer join  
Equi join 
● To join a table itself means that each row of the table is combined with itself and with every other row of the table.
Question 37 
mn  
m+n  
(m+n)/2  
2(m+n) 
If there is common attribute in R and S, and every row of R match with every row of S then total no. of tuples will be mn.
Question 38 
Aggregate Computation  
Multiplication  
Finding transitive closure  
All of the above 
● Multiplication means cartesian product
●Transitive closure:
Given a domain D, let binary relation R be a subset of D×D. The transitive closure R^{ +} of R is the smallest subset of D×D that contains R and satisfies the following condition:
∀ x∀y∀z((x, y ) ∈ R ^{+} ⋀ ( y, z ) ∈ R^{ +} ⇒ ( x, z ) ∈ R ^{+} )
Question 39 
Natural join, outer join  
Outer join, natural join  
Left outer join, right outer join  
Left outer join, natural join 
→ A NATURAL JOIN can be an INNER join, a LEFT OUTER join, or a RIGHT OUTER join. The default is INNER join.
→ The SQL OUTER JOIN returns all rows from both the participating tables which satisfy the join condition along with rows which do not satisfy the join condition. The SQL OUTER JOIN operator (+) is used only on one side of the join condition only.
Question 40 
6  
2  
3  
5 
Consider following two tables
→ Result of natural join R * S (If domain of attribute C in the two tables are same )
→ You can see both R and S contain the attribute C whose value is 2 in each and every tuple. Table R contains 3 tuples, Table S contains 2 tuples, where Result table contains 3*2=6 tuples.
Note: While performing a natural join, if there were no common attributes between the two relations, Natural join will behave as Cartesian Product.
Question 41 
cartesian product always  
combination of union and filtered cartesian product  
combination of selection and filtered cartesian product  
combination of projection and filtered cartesian product 
→ Example: If the tables R and S contains common attributes and value of that attribute in each tuple in both tables are same, then the natural join will result n*m tuples as it will return all combinations of tuples.
Question 42 
Inner Join  
Outer Join  
Semi Join  
Anti Join 
Question 43 
I. ΠA, B (R ⨝ S)
II. R ⨝ ΠB(S)
III. R ∩ (ΠA(R) × ΠB(S))
IV. ΠA, R.B (R × S)
where R⋅B refers to the column B in table R.
One can determine that:
I, III and IV are the same query.  
II, III and IV are the same query.  
I, II and IV are the same query.  
I, II and III are the same query. 
Question 44 
Only the joining attributes are sent from one site to another and then all of the rows are returned.  
All of the attributes are sent from one site to another and then only the required rows are returned.  
Only the joining attributes are sent from one site to another and then only the required rows are returned.  
All of the attributes are sent from one site to another and then only the required rows are returned. 
Question 45 
The number of tuples in the resulting table of RA1, RA2 and RA3 are given by:
2, 4, 2 respectively  
2, 3, 2 respectively  
3, 3, 1 respectively  
3, 4, 1 respectively 
Question 46 
(a)  
(b)  
(c)  
(d) 
Question 47 
(a) join
(b) Intersection
(c) Cartisian product
(d) Project
(a) and (b)  
(b) and (c)  
(c) and (d)  
(b) and (d) 
Question 48 
(a) Relational algebra and Domain relational calculus
(b) Relational algebra and Tuple relational calculus
(c) Relational algebra and Domain relational calculus restricted to safe expression
(d) Relational algebra and Tuple relational calculus restricted to safe expression
(a) and (b) only  
(c) and (d) only  
(a) and (c) only  
(b) and (d) only 
Question 49 
In a system for a restaurant, the main scenario for placing order is given below:
(a) Customer reads menu
(b) Customer places an order
(c) Order is sent to kitchen for preparation
(d) Ordered items are served
(e) Customer requests for a bill for the order
(f) Bill is prepared for this order
(g) Customer is given the bill
(h) Customer pays the bill
A sequence diagram for the scenario will have at least how many objects among whom the messages will be exchanged.
3  
4  
5  
6 
Question 50 
30  
20  
50  
1500 
NATURAL JOIN requires that the two join attributes (or each pair of join attributes) have the same name in both relations.
Let us take a small example where we are having two relations named Employee(EID, Ename) and Department(EID, DID)
Since relation R2 is having 30 tuples, so in best case natural join of R1 and R2 will 30 tuples.
Question 51 
Considering the relation schemas R (A, B, C, D) and S (C. D. E. F), what will be the degree of the resultant relation of the following Relational Algebra expression. Where “*” represents the”natural join" operation?
3  
4  
6  
5 
Question 52 
Consider the following relation schema R and S along with their tuple sets.
R(A, B) = {
S(A) = {a1, a2, a3}
What is the value of TR / S. where "/" represents the Relational Algebra “division" operation?
T(B) = {b1, b3}  
T(B) = {b1, b2, b4}  
T(B) = {b1, b4}  
T(B) = {b1, b3, b4} 
Question 53 
Relations and fields can be renamed in relational algebra using the renaming operator
p
 
σ  
Θ  
⟕

Question 54 
The ____ defines a set of operations on relations, paralleling the usual algebraic operations such as addition, subtraction or multiplication, which operates on numbers.
Relational calculus  
Referential Integrity
 
Relational Algebra  
Relations

Question 55 
Which of the following is a unary operation?
Intersection  
Projection
 
Join  
Cartesian Product

Question 56 
The number of tuples in the result of a left outer join operation will always be
greater than the number of tuples in the result of the corresponding join operation.
 
at least equal to the number of tuples in the result of the corresponding join operation.  
less than the number of tuples in the result of the corresponding join operation.
 
greater than the number of tuples in the result of the corresponding right outer join operation.

Question 57 
R_{1}∪R_{2}  
R_{1}×R_{2}  
R_{1}R_{2}  
R_{1}∩R_{2} 
Question 58 
m+n and O  
mn and O
 
m+n and mn  
mn and m+n 
If every row of r matches with each row of s, i.e., it means, the join attribute has the same value in all the rows of both r and s, then maximum mn tuples possible.
For minimum no. of tuples, if condition of join is not satisfied for any tuple then, minimum 0 tuples will be there.
Question 59 
Outer join  
Inner join  
Natural join  
Self join 
Question 60 
π, ∞, σ  
π, σ, ∞  
σ, π, ∞  
σ, ∞, π 
SELECT
FROM
WHERE
in a sequence.
So it is equivalent to
SELECTπ
FROM ∞
WHEREσ