Part 5: Data Science Application

a) In the context of data science, assume you have a dataset represented

as a set 'D', where each element is a record of individual patient data

including age, gender, and diagnosis. A 'diagnosis' can be represented

as another set 'Diag', where each element is a unique disease.

(i) Define a function 'f' from 'D' to 'Diag' that maps each patient

record to a diagnosis. Discuss the properties of this function and

its implications in the context of data analysis.

(ii) In many real-world datasets, missing values or errors might occur.

These irregularities can be represented as a set 'Err'. Define a

relation 'R' from 'D' to 'Err'. What would be the characteristics

of this relation? How would it be different from a function?

(iii) In a predictive modeling scenario, we often split the dataset into a

training set "Tr' and a testing set "Te'. Can you define a function

from 'D' to 'Tr' and 'Te'? What are the conditions that this

function should satisfy?

(iv) Suppose a new patient's data is represented as a set 'P' = {age,

gender}. A prediction model in machine learning could be seen as

a function that maps 'P' to 'Diag'. Describe how this relates to

the concept of functions in discrete mathematics.

Remember to show your work and explanations using both mathematical

notation and plain English, relating it to the data science context. SageMath

can be used to illustrate some of these concepts if you find it helpful.