Consider a relational schema for storing information related to movies:

Actor$($name$,$ YofB, country$\_$of$\_$birth, bio, gender$)$

Director$($name$,$ YofB, country$\_$of$\_$birth, bio, gender$)$

Producer$($name$,$ YofB, country$\_$of$\_$birth, bio, gender$)$

Crew$($name$,$ YofB, country$\_$of$\_$birth, bio, $)$

Country$($code$,$ country$\_$name, description$)$

Movie$($title$,$ release$\_$year, genre, budget, cost, gross$\_$earnings$)$

DirectorMovie$($d$\_$name, d$\_$YofB, m$\_$title, m$\_$year$)$

ActorMovie$($a$\_$name, a$\_$YofB, m$\_$title, m$\_$year$)$

CrewMovie$($c$\_$name, c$\_$YofB, m$\_$title, m$\_$year$)$

ProducerMovie$($p$\_$name, p$\_$YofB, m$\_$title, m$\_$year, p$\_$type$)$

Foreign key constraints:

$$ Actor$[$country$\_$of$\_$birth$]$ Country$[$code$]$

$$ Director$[$country$\_$of$\_$birth$]$ Country$[$code$]$

$$ Producer$[$country$\_$of$\_$birth$]$ Country$[$code$]$

$$ DirectorMovie$[$d$\_$name, d$\_$YofB$]$ Director$[$name$,$ YofB$]$

$$ DirectorMovie$[$m$\_$title, m$\_$year$]$ Movie$[$name$,$ release$\_$year$]$

$$ ActorMovie$[$a$\_$name, a$\_$YofB$]$ Actor$[$name$,$ YofB$]$

$$ ActorMovie$[$m$\_$title, m$\_$year$]$ Movie$[$name$,$ release$\_$year$]$

$$ ProducerMovie$[$p$\_$name, p$\_$YofB$]$ Producer$[$name$,$ YofB$]$

$$ ProducerMovie$[$m$\_$title, m$\_$year$]$ Movie$[$name$,$ release$\_$year$]$

$$ CrewMovie$[$c$\_$name, c$\_$YofB$]$ Crew$[$name$,$ YofB$]$

$$ CrewMovie$[$m$\_$title, m$\_$year$]$ Movie$[$name$,$ release$\_$year$]$

$2$

In addition:

$$ No attributes are allowed to have NULL values.

$$ Attributes YofB $($year of birth$),$ release$\_$year, budget, cost and gross$\_$earnings have integer values,

e$.$g$.,1978,2023,1000234.$

$$ All other attributes have strings as values. Attribute code has as values strings such as$,$CA$,$

$$GBR$.$ Attribute genre in table Movie has as values one of $\{$comedy$,$drama$,$tragedy$,$

$$musical$,$horror$\}.$ Attribute p$\_$type in table ProducerMovie has as values one of $\{$executive

producer$,$assistant producer$,$producer$\}.$

Part $1.$Write Relational Algebra expressions for the following queries. Do not use extended Relational Algebra operators, e$.$g$.,$ grouping,

counting, etc.

a$.$ List the name and year of birth of all those who acted in at least two movies which they also

directed.

b$.$ List the name and year of birth of all directors who were younger than all the actors cast in the

movies they directed, at the time the movie was released. $($The persons returned by this query may

be associated with movies in other capacities as well, e$.$g$.,$ as actors, directors, executive

producers, etc. Person A is $$younger$$ than person B is the age of person A is less or equal to the

age of person B$.)$

c$.$ List the name and year of birth of all producers who only produced in one type of movie, for

example only acted in comedies. $($The persons returned by this query may be associated with

movies in other capacities as well, e$.$g$.,$ as actors, directors, etc.$)$