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.

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