The division operator of relational algebra, “÷”, is defined as follows. Let r(R)and s(S) be relations, and let S ⊆ R; that is, every attribute of schema S isalso in schema R. Given a tuple t, let t[S] denote the projection of tuple t onthe attributes in S. Then r ÷ s is a relation on schema R − S (that is, on theschema containing all attributes of schema R that are not in schema S). A tuplet is in r ÷ s if and only if both of two conditions hold:• t is in ΠR−S(r)• For every tuple ts in s, there is a tuple tr in r satisfying both of the following:a. tr[S] = ts[S]b. tr[R − S] = tGiven the above definition:a. Write a relational algebra expression using the division operator to findthe IDs of all students who have taken all Comp. Sci. courses. (Hint:project takes to just ID and course_id, and generate the set of all Comp.Sci. course_ids using a select expression, before doing the division.)b. Show how to write the above query in relational algebra, without usingdivision. (By doing so, you would have shown how to define the divisionoperation using the other relational algebra operations.)
The division operator of relational algebra, “÷”, is defined as follows. Let r(R)
and s(S) be relations, and let S ⊆ R; that is, every attribute of schema S is
also in schema R. Given a tuple t, let t[S] denote the projection of tuple t on
the attributes in S. Then r ÷ s is a relation on schema R − S (that is, on the
schema containing all attributes of schema R that are not in schema S). A tuple
t is in r ÷ s if and only if both of two conditions hold:
• t is in ΠR−S(r)
• For every tuple ts in s, there is a tuple tr in r satisfying both of the following:
a. tr[S] = ts[S]
b. tr[R − S] = t
Given the above definition:
a. Write a relational algebra expression using the division operator to find
the IDs of all students who have taken all Comp. Sci. courses. (Hint:
project takes to just ID and course_id, and generate the set of all Comp.
Sci. course_ids using a select expression, before doing the division.)
b. Show how to write the above query in relational algebra, without using
division. (By doing so, you would have shown how to define the division
operation using the other relational algebra operations.)
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