$(/8$ marks$)$ Write the following queries in relational algebra, using the university schema $($see appendix A$.1$ and A$.3$ in your textbook for the University schema and instance respectively$).$ Hint: see Lesson $02$ presentation, slide $45$ "Approach to writing relational algebra queries", for approach on writing relational algebra queries and make sure if you have multiple tables that you compute the joins correctly $($watch out for name collisions if using natural join$).$

a$.$ Find the names of all students who have taken at least one Comp. Sci. course $(2$ marks$).$ Hint: Start by figuring out to list all the courses taught by the computer science department.

b$.$ Find the IDs and names of all students who have not taken any course offering before Spring $2017(2$ marks$).$ Hint: assume that spring is the first semester of the year when writing your comparison expression

c$.$ For each department, find the maximum salary of instructors in that department. You may assume that every department has at least one instructor $(2$ marks$).$ Hint: be careful to use the correct syntax for aggregate functions, as per the lecture slides.

d$.$ Find the lowest, across all departments, of the per$-$department maximum salary computed by the preceding query $(2$ marks$).$ Hint: figure out how to calculate out the maximum salaries for each department first.Figure A$.3$ The department relation.

Figure A$.1$ Schema diagram for the university database.