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+25 votes

Consider the following relational database schemes:

  • COURSES (Cno, Name)
  • PRE_REQ(Cno, Pre_Cno)
  • COMPLETED (Student_no, Cno)

COURSES gives the number and name of all the available courses.

PRE_REQ gives the information about which courses are pre-requisites for a given course.

COMPLETED indicates what courses have been completed by students

Express the following using relational algebra:

List all the courses for which a student with Student_no 2310 has completed all the pre-requisites.

in Databases by Veteran (52.3k points)
edited by | 2.5k views

In relation COURSES  ,(comma) missing

 COURSES ( -> COURSES (Cno. , name)

see detail explanation at the last of this page...

check queries on this tool, it's awesome.
$\text{COMPLETED-COURSES} \leftarrow \rho \ (\sigma_{(student\_no = 2310)} \text{COMPLETED})$


$\pi_{\text{PRE-REQ.Cno}} (\text{COMPLETED-COURSES}\ \boxtimes \ _{\text{(COMPLETED-COURSES.Cno} = \text{PRE-REQ.pre-Cno)}} \text{PRE-REQ}) $


Is this relational algebra expression correct?

10 Answers

+13 votes
Best answer

$T_1$ will have all the available course numbers

$T_2$ will have all the course numbers completed by student2310

$T_3$  will have the combination of all the courses and the courses completed by student2310

$\text{PRE_REQ} - T_3$ (set minus operation) will return us all the entries of $\text{PRE_REQ}$ which are not there in $T_3,$

Suppose $\langle C_1,C_5\rangle$ is a particular tuple of $(\text{PRE-REQ} - T_3),$

Now what does it imply? $\implies$ It implies that $C_5$ is one of the prerequisite course for $C_1$ which has not been completed by $C_5$. Proof: If student2310 would have completed $C_5$ then definitely $\langle C_1,C_5 \rangle$ should have been there in  $T_3$ (remember $T_3$ is the combination of all the courses and the courses completed by student2310) and in that case $(\text{PRE_REQ} - T_3)$ can not have  $\langle C_1,C_5 \rangle$ as a tuple.

So, for any such $\langle C_1,C_5 \rangle$ tuple, $(\langle C_1,$ any  course id$\rangle)$ of $\text{PRE_REQ} - T_3, C_1$  should not be printed as output (Since there is some prerequisite course for $C_1$ which student2310 has not completed).

Now, suppose we have not got any tuple as a result of $(\text{PRE_REQ} - T_3)$ where $C_2$ is there under cno attribute $(\langle C_2,$ any course id$\rangle ),$ what does it imply?$\implies $ It implies that student2310 has completed all the prerequisite courses $C_2.$

Hence, in order to get the final result we need to project cno from $(\text{PRE_REQ} - T_3)$ and subtract it from $T_1.$

  • $T_1 \leftarrow \pi_{\text{cno}}(\text{COURSES})$
  • $T_2 \leftarrow \rho_{T_2(\text{std2310completedcourses})}(\pi_{\text{cno}}(\sigma_{\text{student_no} = 2310}(\text{COMPLETED})))$
  • $T_3 \leftarrow T_1 \times T_2$
  • $T_4 \leftarrow \rho_{T_4(\text{cno, pre_cno})}(\text{PRE_REQ}-T_3)$
  • $Result \leftarrow T_1 - \pi_{\text{cno}}(T_4)$
by Active (3k points)
edited by
this is the correct answer. Rest all above give wrong answer.

This seems the best answer. @Arjun sir please check this.

I've always one doubt whenever see this type of questions.How to think like this in exam hall within a time limit.

+11 votes
$S \leftarrow \pi_{Cno}(\sigma_{student\_no=2310}(COMPLETED))\\RESULT \leftarrow ((\rho_{(Course,Cno)}(PRE-REQ)) \div S )$
by Boss (18k points)

we will select CNO for stud_no=2310 and rename CNO as pre-cno in COMPLETED relation and then perform divison of PreReq table with the table created above. Right??
No, it won't work. Say if student completed two courses C1 and C2 that are prerequisite for C3 and C4 respectively. Then what will ur query return?. It will return Null table.
Bcoz neither C3 has both prerequisite as C1 and C2 nor C4 has both prerequisite.
so can u write the correct query in relational algebra as sql both??


The division operation p(P) ÷ q(Q) only makes sense when Q ⊆ P.

In the above assignment, there are no common attributes between the two relations as the only common attribute 'Cno' has been renamed to 'Course', so the division makes no sense.

Your answer is incorrect because it is not even a valid expression.

Can I do this

Project courseno (Select studentno=2310 (completed join pre requisite))

Will this work for the below database?


Cno Cname
c1 A
c2 B
c3 C
c4 D
c5 E


Cno Pre-Cno
c1 c3
c2 c1
c2 c4
c5 c2
c5 c3

In graphical form:


Which shows that c3 and c4 do not need any pre-course and rest others do. There shouldn't be any cycle in this graph otherwise no course can ever be completed. Like if pre-course for c1 is c2 and pre-course for c2 is c1 then we cannot make any progress.

Sno Cno
s1 c3
s1 c4
s2 c1
s3 c2

Now if I want to know what are the courses that s1 can complete after having taken the pre-courses, then:

pre courses taken by s1 are c3 and c4.

With the knowledge of c3 and c4 only c1 can be completed. 

After taking c1, s1 can then take c2 and then c5. But these shouldn't be the current result.

According to the query:

S will have {c3,c4} and when we divide pre-req with it, the result we get is null. Isn't it? If so then this is not giving correct result. 

@Sachin I saw the pdf you attached.. But there you haven't considered the case if a student has completed more than one course. 

Please help :/



R←πCnoCno X S

M←πCno(Pre-Req - R)

πCno(πCno(Cno) - (M∪S) )

I am getting this.

For S1,


R= <c1,c3>, <c1,c4>,<c2,c3>,<c2,c4>,<c3,c3>,<c3,c4>,<c4,c3>,<c4,c4>,<c5,c3>,<c5,c4>

M= πCno(<c2,c1>,<c5,c2>) = {c2,c5}

πCno(Cno - (M∪S) )= πCno({c1,c2,c3,c4,c5} - ( {c2,c5}∪{c3,c4}) ) = c1

For S2, answer should be c3 and c4 only because they don't need any pre-course.

Please correct me if wrong.


 yes i also think in the same way.  answer seem to be same as yours.


@Arjun Sir, Please check this answer

yes, this is wrong. It is outputting the courses only if the student has not completed any other courses than its prerequisites.
+7 votes

SQL query will be 

FROM Completed, Pre-Req  
WHERE student_no = '2310'  
GROUP BY cno  
HAVING pre-Cno IN (
    SELECT C.cno 
    FROM Completed AS C
    WHERE C.student_no = '2310';
by Boss (30.8k points)
But question is for relational algebra query.
arjun sir can u plz write RELATIONAL ALGEBRA for it?? i am not able to figure out. :-(

The one given by @mint below looks right to me. Verify
ΠCno.(PRE-REQ ⋈pre Cno.=Cno.   ΠCno.Sno.=2310(COMPLETED)))

Is it right sql query for above scenario???

select COURSES.cno

@amarVashishth I think your sql query won't work because "IN" operator is a shorthand for multiple "OR" conditions in SQL so here any course no will be displayed for which any one of the pre-req course has been completed But here we need For All condition.

FROM Courses C1
                                       FROM PRE-REQ P1
                                       WHERE C1.Cno=P1.Cno
                                       SELECT Cno
                                       FROM COMPLETED C2
                                       WHERE C2.student_no='2310' ) ) ;

+4 votes

RA query: ΠCno.(PRE-REQ ⋈pre Cno.=Cno.   ΠCno.Sno.=2310(COMPLETED)))   

make it in two parts:

X= ΠCno.Sno.=2310(COMPLETED))      {X gives all the tuples in which 2310 is present}

YCno.(PRE-REQ pre Cno.=Cno.   X)   

I think Y is the required answer... correct me if i wrong.

by Active (1.5k points)
I got the same answer. I am wondering my no one commented here before. Veterans ?

PRE-REQ gives the information about which courses are pre-requisites for a given course, hence a course may have many prerequisite courses. In that case suppose a course C1 has two prerequisite courses- CP1 and CP2, now even if Sno-2310 completes only one of those prerequisite courses then also above query will print C1, which it should not, if and only if Sno-2310 completes both CP1 and CP2 then only it should print C1

There maybe more than one prereq for a single course.
+1 vote

FROM Courses C1


                                      FROM PRE-REQ P1

                                      WHERE C1.Cno=P1.Cno


                                      SELECT Cno

                                      FROM COMPLETED C2

                                      WHERE C2.student_no='2310' ) ) ;

Is this query correct ?
by (51 points)
reshown by
+1 vote

I came up with two approach for this Question.

Note- Questions in which "ALL" keyword is used, most probably would be solved with division operation.  

by Active (4.5k points)
0 votes

This should be the answer.

ΠCno(PRE-REQ) - ΠCno(PRE-REQ - ρPRE-REQ.Cno/CnoPRE-REQ.Cno,pre-Cno((σstudent_no=2310 (COMPLETED))⋈Cno=pre-Cno PRE-REQ)))

by Active (1.8k points)
edited by
0 votes

Correct me If I am Wrong..

Correct me if I am Wrong..

by (261 points)
edited by

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