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The problem involves interpreting the data from a survey about diets and supplements, as well as solving set operations related to the data.

Here’s a breakdown of the data from the table:

### Diets:
- \( Z \): Zone Diet
- \( S \): South Beach Diet
- \( W \): Weight Watchers Diet

### Supplements:
- \( E \): Exercise Program
- \( P \): Diet Pills
- \( M \): Medical Supervision

### Table Data:

| Diet | E  | P  | M  | Totals |
|------|----|----|----|--------|
| Z    | 124| 82 | 65 | 271    |
| S    | 101| 66 | 51 | 218    |
| W    | 133| 41 | 48 | 222    |
| Totals | 358| 189| 164| 711    |

You are asked to find the number of surveyed people for the following sets:

**a. \( S \cap E \):**  
People on the South Beach Diet (\(S\)) who are also in an Exercise Program (\(E\)):  
This value is directly given in the table: 101 people.

**b. \( Z \cup M \):**  
People either on the Zone Diet (\(Z\)) or under Medical Supervision (\(M\)) or both.  
We need to count those under the Zone Diet (total: 271), those under Medical Supervision (total: 164), and subtract those double-counted (Zone Diet under Medical Supervision, \(Z \cap M\)): 65.  
Thus, \( |Z \cup M| = 271 + 164 - 65 = 370 \).

**c. \( S' \cap (E \cup P) \):**  
People not on the South Beach Diet (\(S'\)) but either in an Exercise Program (\(E\)) or taking Diet Pills (\(P\)) or both.  
First, find people not in \(S\), which includes people from \(Z\) and \(W\) diets:
- \( Z \cap (E \cup P) = 124 + 82 = 206 \) 
- \( W \cap (E \cup P) = 133 + 41 = 174 \)  
So, \( |S' \cap (E \cup P)| = 206 + 174 = 380 \).

**d. \( (Z \cup S) \cap M' \):**  
People either on the Zone Diet (\(Z\)) or South Beach Diet (\(S\)) who are not under Medical Supervision (\(M'\)).  
From the table:
- Total people on \( Z \cup S \) is \( 271 + 218 = 489 \).
- People in \( Z \cap M \) (Zone Diet under Medical Supervision) = 65
- People in \( S \cap M \) (South Beach Diet under Medical Supervision) = 51  
Thus, \( |(Z \cup S) \cap M'| = 489 - (65 + 51) = 373 \).

**e. \( W' \cap (P \cup M) \):**  
People not on the Weight Watchers Diet (\(W'\)) but either taking Diet Pills (\(P\)) or under Medical Supervision (\(M\)) or both.  
People on \( Z \cap (P \cup M) = 82 + 65 = 147 \).  
People on \( S \cap (P \cup M) = 66 + 51 = 117 \).  
So, \( |W' \cap (P \cup M)| = 147 + 117 = 264 \).

**f. \( W' \cup P \):**  
People either not on Weight Watchers (\(W'\)) or taking Diet Pills (\(P\)) or both.  
People not on Weight Watchers include those from \( Z \) and \( S \):
- Total for \( Z \cup S \): \( 271 + 218 = 489 \).
People taking Diet Pills \(P\): \( 189 \).  
We subtract the overlap, people in \( Z \cup S \) who also take diet pills, \( |P \cap (Z \cup S)| = 82 + 66 = 148 \).  
Thus, \( |W' \cup P| = 489 + 189 - 148 = 530 \).

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Would you like more details on any of these answers?  
Here are 5 related questions for further practice:
1. How many people were on the South Beach Diet and under Medical Supervision?
2. What is the total number of people who participated in an Exercise Program across all diets?
3. How many people were not taking Diet Pills but were either in an Exercise Program or under Medical Supervision?
4. How many people were on the Weight Watchers Diet and not under Medical Supervision?
5. How many people were taking Diet Pills or on the Zone Diet?

**Tip:** When working with set operations like unions and intersections, always remember to subtract the overlap to avoid double counting.

Find the number of surveyed people in each of the following sets:
a. S ∩ E
b. Z ∪ M
c. S' ∩ (E ∪ P)
d. (Z ∪ S) ∩ M'
e. W' ∩ (P ∪ M)
f. W' ∪ P

Solve a problem involving set operations in a dietary survey. Learn how to calculate unions, intersections, and complements using data about diets and supplements, applying key set theory concepts.

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