A library management system is used to store information about books. Each book can

be associated with a set of tags. The system allows users to search for books by tag.

Consider the following book collection $C$ which has $5$ books: $C=\{B_1,B_2,B_3,B_4,B_5\}.$

Let Dict be a dictionary which consists of $6$ tags: Dict $=\{\begin{array}{c}{t}_1=\end{array}$ Science Fiction, $t_2=$

Adventure, $t_3=$ Non$-$Fiction$\}.$

a$)$ Denote by $tf(t,M)$ the tag frequency $($TF$)$ of the tag $t$ in the book $B.$ Please fill out the

blank cells in the following table, i$.$e$.,$ give the values $tf(t_i{B}_j).$ The value $tf(t_i{B}_j)$ should

be put into the cell specified by $t_i$ and $M_j.($Note$,$ you can copy the form to your answer

sheet and the fill it out.$)$

b$)$ Recall the inverse document frequency $($IDF$)$ is defined as idf$(t,C)=ln(\frac{|C|}{|C_t|}).$ Here $|C|$

denotes the number of books in the collection $C,|C_t|$ denotes the number of books from

$C$ that contains tag $t.$ Please compute idf$(t_1,C)$ and idf$(t_3,C).$ c$)$ Tag frequency $$ inverse document frequency $($TF$-$IDF$)$ takes both tag frequency $($TF$)$ and inverse document frequency $($IDF$)$ into consideration. For the book collection $,$ the tag $$s TF$-$IDF value on book is defined as $,$ i$.$e$.,$ the product of $$s TF value on and $$s IDF value. Please compute and $.$

d$)$ Using TF$-$IDF, a book can be represented by a multi$-$dimension vector of TF$-$IDF values of all tags in the dictionary. Compute the vector for each book.

e$)$ Similarly, any given query can be represented by a multi$-$dimension vector. Compute the vector for the query $$Science Fiction$.$

f$)$ Using document vectors, we can compute the relevance score for each book to a given query using cosine similarity between the book vector and the query vector. A retrieval result of a query is the ranking of the books in the decreasing order of relevance scores. What is the result of the query $$Science Fiction$?$

g$)$ What is the result of the query $$Science$?$ Explain why. Which solution do you suggest to include B$2$ and B$3$ in the result.