$2.$ Consider the following relational schema.

Emp$($eid: integer, ename: string, age: integer, salary: real$)$

Works$($eid: integer, did: integer, pct$\_$time: integer$)$

Dept$($did: integer, dname: string, budget: real, managerid: integer$)$

Write the following queries in relational algebra.

a$.$ Print the names and ages of each employee who works in both the Hardware department and the Software department. $(5$ points$)$

b$.$ Print the name of each employee whose salary exceeds the budget of all the departments that he or she works in$.$ To illustrate, if the employee works in $3$ departments, whose budgets are $1000,2000$ and $3000$ respectively, then the maximum budget of all the departments that he$/$she works in is $3000.(5$ points$)$

c$.$ Find the ename, age of managers who manage only departments with budgets greater than $\backslash (\backslash$$ $1\backslash )$ million. $(5$ points$)$

d$.$ Find the enames of managers who manage the departments with the greatest budgets. To illustrate, suppose there are three departments like this: department $1$ with a budget of $\backslash (\backslash$$ $1\backslash )$ million, department $2$ with a budget of $\backslash (\backslash$$ $2\backslash )$ million, department $3$ with a budget of $\backslash (\backslash$$ $2\backslash )$ million. Then departments $2$ and $3$ have the greatest budgets. $(9$ points$)$

Part II$.$ SQL Queries. $(30$ points$)$

$-$ Copy all SQL queries into one sql file called part$2.$sql$.$ For each query, please provide the comment as "part$2$a$"$ for the first query.

$-$ All screenshots should be submitted together as a single pdf file called part$2.$pdf$.$

$-$ Upload your submissions as a single archive file called part$2.$zip.

Consider the relational schema for company in Part I$\mathrm{.}2,$ write SQL queries for each request in Part I. $2$ and one extra request in the following. Your queries should be executable in SQLite.

In this part, you also need to use the $"$A$2.$db$"($please download $"$A$2.$db$"$ from course files$)$ database in DB Browser for SQLite to give a screenshot for each query's result.

a$.6$ points

b$.5$ points

c$.5$ points

d$.6$ points

e$.$ For each department with more than $20$ full$-$time$-$equivalent employees $($i$.$e$.,$ the total pct$\_$time of the part$-$time and full$-$time employees add up to at least $2000$ hrs$,$ assuming that the pct$\_$time of a full$-$time employee is $100$ hrs $),$ print the did together with the number of employees that work in that department. $(8$ points$)$