Exercise $2\mathrm{.}4\mathrm{.}3$: This exercise builds upon Exercise $2\mathrm{.}3\mathrm{.}2$ concerning World

War II capital ships. Recall it involves the following relations:

Classes $($class$,$ type, country, numGuns, bore, displacement$)$

Ships $($name$,$ class, launched$)$

Battles $($name$,$ date$)$

Outcomes $($ship$,$ battle, result$)$

Figures $2\mathrm{.}22$ and $2\mathrm{.}23$ give some sample data for these four relations. ${?}^4$ Note

that, unlike the data for Exercise $2\mathrm{.}4\mathrm{.}1,$ there are some "dangling tuples" in this

data, e$.$g$.,$ ships mentioned in Outcomes that are not mentioned in Ships.

Write expressions of relational algebra to answer the following queries. You

may use the linear notation of Section $2\mathrm{.}4\mathrm{.}13$ if you wish. For the data of Figs.

$2\mathrm{.}22$ and $2\mathrm{.}23,$ show the result of your query. However, your answer should work

for arbitrary data, not just the data of these figures.

a$)$ Give the class names and countries of the classes that carried guns of at

least $16-$inch bore.

b$)$ Find the ships launched prior to $1921.$

c$)$ Find the ships sunk in the battle of the Denmark Strait.

d$)$ The treaty of Washington in $1921$ prohibited capital ships heavier than

$35,000$ tons. List the ships that violated the treaty of Washington.

e$)$ List the name, displacement, and number of guns of the ships engaged in

the battle of Guadalcanal.

f$)$ List all the capital ships mentioned in the database. $($Remember that all

these ships may not appear in the Ships relation.$)$

use expression trees to solve f$.$