Part $4$: Relations and Functions

Imagine you are tasked with designing a smart library system that involves mapping books to categories, authors to genres and customers to book preferences. Your goal is to ensure that the library system is efficient and can accurately reflect relationships between books, authors and patrons.

a$)$ Construct the specific sets provided below:

$$ Books $($B$)$: Create a set of $10$ books with unique titles.

$$ Categories $($C$)$: Define a set of $5$ book categories $($e$.$g$.,$ Fiction, Non Fiction, Science, History, Fantasy$).$

$$ Authors $($A$)$: Define a set of $5$ authors with unique names.

$$ Genres $($G$)$: Create a set of $3$ genres.

$$ Patrons $($P$)$: Define a set of $6$ library patrons.

b$)$ Define and describe the functions and relations provided below. Include any necessary diagrams or tables:

$$ Function f: Define a function f that maps each book to one or more categories. Discuss how this function ensures that each book is categorized appropriately.

$$ Relation r: Define a relation r that maps each author to one or more genres. Describe the characteristics of this relation and how it differs from a function.

$$ Function g: Define a function g that maps each patron to a list of their preferred books. Explain how this function can be used to recommend books to patrons based on their preferences.

c$)$ Gather the following data:

$$ Books and Categories: Create a physical or digital list of books and assign each book to one or more categories. For instance, "Book$1"$ might belong to "Fiction" and "Fantasy".

$$ Authors and Genres: Similarly, list the authors and assign each author to one or more genres. For example, "Author$1"$ could be associated with "Science" and "History".

$$ Patron Preferences: List each patron and their preferred books. For example, "Patron$1"$ might prefer "Book$2"$ and "Book$3".$

d$)$ Evaluate if f is a well$-$defined function. If a book can belong to multiple categories, how does this affect the function's definition? Are there any cases where a book might not fit into any category?

e$)$ Examine the function g$.$ Discuss how this function helps in personalizing book recommendations and any limitations it might have.

$3$

f$)$ Discuss the relation r between authors and genres. Is it a function? Why or why not? How would you handle cases where an author writes across multiple genres?

g$)$ Consider scenarios where patrons have overlapping preferences or where books fall into multiple categories. How would you adjust the system to accommodate these overlaps and ensure effective recommendations?

h$)$ Discuss any potential limitations and improvements to the mapping system. Based on the data collected, propose enhancements to improve the system. How would you increase the accuracy of book categorization, author$-$genre associations and patron preferences?