did I do this correctly relational algebraic expressions for the following queries and how would their query trees look like.

Consider the following relational schema and briefly answer the questions that follow. Make any assumptions ad state them as part of your answer.

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

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

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

$1.$ Write relational algebraic expressions for the following queries and draw their query trees.

a$)$ Retrieve the names of department where $$John Smith$$ works.

Emp$\_$JS $<$ ename$=$John$$Smith$($Emp$)$

Dept$\_$JS $<$ did $($Works $$ eid Emp$\_$JS$)$

Deptnames$\_$result $<$ dname$($Dept $$ did Dept$\_$JS$)$

b$)$ List all employees who work for a manager younger than themselves.

ManagerInfo $<$ Emp AS E$1$ managerid Dept

Manager$\_$EmpInfo $<$ ManagerInfo $$ eid $=$ Works.eid Emp AS E$2$

Younger$\_$Mangers $<$ ename$($E$2.$age $>$ E$1.$age$($Manager$\_$EmpInfo$))$

c$)$ List all employees who do not work in any department managed by $$Jane Smith$.$

Emp$\_$JaneS $<$ ename $=$ 'Jane Smith'$($Emp$)$

Dept$\_$JaneS $=$ did $($Dept $$ managerid Emp$\_$JaneS$)$

Emp$\_$JaneSDept $=$ eid$($Works $$ did Dept$\_$JaneS$)$

ExcludedEmployees $=$ eid$($Emp$)-$ Emp$\_$JaneSDept

d$)$ List all employees who work in every department managed by $$Jane Smith$.$

Emp$\_$JaneS $<$ ename $=$Jane Smith'$($Emp$)$

Dept$\_$JaneS $<$ did$($Dept $$managerid Emp$\_$JaneS$)$

Emp$\_$JaneSDepts $=$ eid$($Works $$ did Dept$\_$JaneS$)$

Employees$\_$AllDepts $=$ eid$($Emp$)$ did$($Dept$\_$JaneS$)$

e$)$ List all departments were everyone who works is above $50$ years old.

DeptEmp $<$ Works $$ eid Emp

DeptOlderEmp $<$ did$($age $>50($DeptEmp$))$

f$)$ Find managers who manage more than one department.

DeptManagers $=$ managerid$,$ count$($did$)($Dept$)$

MultiDeptManagers $=$ count$($did$)>1($DeptManagers$)$

g$)$ Find a manager who works more than any of his$/$her employees in any department.

ManagerDetails $=$ Dept $$managerid Emp AS Manager

EmployeeDetails $=$ Works $$eid Emp AS Employee

ManagersWithHigherWork $=$ managerid$($Manager$.$salary $>$ Employee.salary$($ManagerDetails $$ EmployeeDetails$))$

h$)$ Find total hours worked by all employees in department number $5.$

Dept$5$Emp $=$ did$=5($Works$)$

TotalHoursDept$5=$ SUM$($pct$\_$time$)($Dept$5$Emp$)$