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 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…

The following digraph represents a relation R. a C b d List the ordered pairs in the relation. R = { }.

Suppose we have the following relation schema R and set of functional dependencies F: R = (A, B, C, D, E, F, G) F = { A->BC CD - AB EF AD В - АЕF} For each of the following, determine whether it is in F+ (i.e., can be derived from F). If the answer is yes, show how it can be derived step by step (and indicate which rule in Armstrong axioms is involved). If not, explain why. (а) AD-> EF (b) B CD (c) DE → AC (d) CEF – G