١٠/ SECTION A Let the universal set Ube the set of all patients in a clinical study. We define the following sets: A= (x e Uix iias Hypertension), B= (x e Ulx has Diabetes). = (x E Ulx has High Cholestro!) Describe the set AUB in a single sentence. What medical condition does this group represent? (ii) A new set D = [x E U|x > 65 years old} is defined. What does the set (AU B) n D represent in terms of public health planning? (iii) (iv) Using words, describe the patients in the set B'. Describe the set A-C. What is the key difference between the patients in A- Cand those in An C*? (V) A doctor is interested in patients who have Diabetes but do not have the comorbidity of (vi) (vil) High Cholesterol. Write this using set notation. Deline the symmetric difference A 4 B in words. What does |/ A B| represent? Using the set notation, express the set of patients who have exactly one of the three conditions: Flypertension, Diabetes, or High Cholesterol. (Vill) Let D= (Fever, Cough) represent a minimal set of symptoms for a disease. List the power set of(D)

let's tackle this set theory problem.

a)

(i) Describe the set A U B in a single sentence. What medical condition does this group represent?
The set A U B represents patients who have Hypertension or Diabetes (or both). This group represents patients with at least one of these two chronic medical conditions.

(ii) A new set D = {x $\in$ U | x > 65 years old} is defined. What does the set (A U B) $\cap$ D represent in terms of public health planning?
The set (A U B) $\cap$ D represents patients who are over 65 years old and have either Hypertension or Diabetes (or both). In public health planning, this group signifies elderly patients with specific chronic conditions, which is crucial for targeted healthcare interventions, resource allocation, and specialized geriatric care programs.

(iii) Using words, describe the patients in the set B'.
The patients in the set B' are those who do not have Diabetes.

(iv) Describe the set A - C. What is the key difference between the patients in A - C and those in A $\cap$ C'?
The set A - C describes patients who have Hypertension but do not have High Cholesterol.
There is no key difference between the patients in A - C and those in A $\cap$ C'. These two notations represent the exact same set of patients: those who have Hypertension and simultaneously do not have High Cholesterol. The notation A - C is equivalent to A $\cap$ C'.

(v) A doctor is interested in patients who have Diabetes but do not have the comorbidity of High Cholesterol. Write this using set notation.

$$B \cap C'$$

or

$$B - C$$

(vi) Define the symmetric difference A $\Delta$ B in words. What does $|A \Delta B|$ represent?
The symmetric difference A $\Delta$ B is the set of patients who have Hypertension or Diabetes, but not both. It includes patients who have Hypertension only, or Diabetes only.
$|A \Delta B|$ represents the number of patients who have exactly one of the two conditions (Hypertension or Diabetes).

(vii) Using the set notation, express the set of patients who have exactly one of the three conditions: Hypertension, Diabetes, or High Cholesterol.

$$(A \cap B' \cap C') \cup (B \cap A' \cap C') \cup (C \cap A' \cap B')$$

(viii) Let D = {Fever, Cough} represent a minimal set of symptoms for a disease. List the power set of (D).
The power set of D is:

$$\boxed{\{\emptyset, \{\text{Fever}\}, \{\text{Cough}\}, \{\text{Fever, Cough}\}\}}$$