$(24$ points$)$ Consider the following relations, where the primary keys are underlined.

$$ Customer $($cid$,$ cname, age$)$

$$ Reserve $($cid$,$ hid, no$,$ date$)$

$$ Hotel $($hid$,$ hname$)$

$$ Room $($hid$,$ no$,$ rate$)$

Let C be Customer, let RE be Reserve, let R be Room, and let H be Hotel.

Write the relational algebra for the following queries.

$1)(3$ points$)$ Retrieve the names of customers who have reserved at least one room in the $$FAU$$

hotel.

$2)(3$ points$)$ Retrieve the names of hotels where a room rate $($or price$)$ is higher than $$1,000.$

$3)(3$ points$)$ Retrieve the names of customers who have never reserved a room in the $$FAU$$ hotel.

$4)(3$ points$)$ Retrieve the names of customers who have made reservations at either the 'FAU' hotel

or the 'Boca' hotel.

$5)(3$ points$)$ Retrieve the names of customers who have made reservations at both the 'FAU' hotel

and the 'Boca' hotel.

$6)(3$ points$)$ Retrieve the names of customers who have reserved the same room on different dates.

$7)(3$ points$)$ Retrieve the names of customers who have reserved all rooms in the $$FAU$$ hotel.

$8)(3$ points$)($Do not use the aggregation function$)$ Retrieve the name of the hotel with the highest

room rate $($or price$).$