4. What is the difference between the natural join RS and the theta-join RC S where the condition C is that R.A = S.A for each attribute A appearing in the schemas of both R and S? For example, if the relations are R(A, B, C, D) and S(C, D, E), what is the difference between R S and RR.C-S.C AND R.D=S.D S?

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…

Faculty (facultyNum, deptNum, officeNum, rank, dateHired, salary) ● Where: officeNum deptNum rank, dateHired → salary What is the highest normal form that the above relation satisfies? Decomposition into BCNF:

Make a hybrid merge-join variant that works in scenarios where neither relation is physically sorted but has a sorted secondary index on the joined attributes.