staff $=\{$uid$,$ name$\}$

primary key: $\{$uid$\}$

course $=\{$code$,$ name, level$\}$

primary key: $\{$ccode$\}$

teach $=\{$uid$,$ ccode, sem, year, role$\}$

primary key: $\{$uid$,$ ccode, sem, year$\}$

foreign key: $[$uid$]$ staff$[$uid$]$

foreign key: $[$ccode$]$ course$[$ccode$]$ and query: select distinct tutor.name

from $($staff join teach on $($staff$.$uid $=$ teach.uid$))$ as tutor,

$($staff join teach on $($staff$.$uid $=$ teach.uid$))$ as convener, course

where tutor.ccode $=$ convener.ccode

and tutor.sem $=$ convener.sem

and tutor.year $=$ convener.year

and convener.role $=$ 'convener'

and tutor.role $=$ 'tutor'

and convener.ccode $=$ course.ccode

and convener.name $=$ 'Graham'

and course.level $=2000$;

$1.$ Translate the above query into a relational algebra query, using only the operators in the table above. You may use relational algebra expressions to represent intermediate results if needed. $2$ Draw the query tree for the relational algebra query you constructed for question $(1).3.$ Optimise the relational algebra query.