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I can help you analyze the math problem based on the image. It appears to involve set theory with different categories of diets and supplements. The problem asks for the number of people surveyed in specific sets, namely:

1. \( S' \cap (E \cup P) \)
2. \( W' \cap (P \cup M) \)

Let’s break it down:

### 1. \( S' \cap (E \cup P) \)
This is asking for the people who are not on the South Beach diet (\( S' \)) but are either in the exercise program (\( E \)) or taking diet pills (\( P \)).

From the table:
- \( S \) represents South Beach diet people, and \( S' \) refers to people **not** on the South Beach diet, so this includes the Zone (\( Z \)) and Weight Watchers (\( W \)) groups.
- \( E \cup P \) means the union of people in exercise programs (\( E \)) or taking diet pills (\( P \)).

Thus, we need to count people in \( Z \) and \( W \) rows, under the \( E \) and \( P \) columns:
- From \( Z \): \( 123 \) (exercise) + \( 81 \) (diet pills) = \( 204 \)
- From \( W \): \( 132 \) (exercise) + \( 41 \) (diet pills) = \( 173 \)

Total for \( S' \cap (E \cup P) = 204 + 173 = 377 \).

### 2. \( W' \cap (P \cup M) \)
This is asking for the people who are not on the Weight Watchers diet (\( W' \)) but are either taking diet pills (\( P \)) or under medical supervision (\( M \)).

From the table:
- \( W \) represents Weight Watchers diet people, so \( W' \) refers to people **not** on the Weight Watchers diet, which includes the Zone (\( Z \)) and South Beach (\( S \)) groups.
- \( P \cup M \) means the union of people taking diet pills (\( P \)) or under medical supervision (\( M \)).

Thus, we need to count people in \( Z \) and \( S \) rows, under the \( P \) and \( M \) columns:
- From \( Z \): \( 81 \) (diet pills) + \( 65 \) (medical supervision) = \( 146 \)
- From \( S \): \( 67 \) (diet pills) + \( 51 \) (medical supervision) = \( 118 \)

Total for \( W' \cap (P \cup M) = 146 + 118 = 264 \).

If you need more details or have any questions, feel free to ask! 

Here are 5 related questions to think about:
1. How would the result change if we considered \( S' \cap (E \cap P) \)?
2. What would be the result of \( W' \cap (E \cup P \cup M) \)?
3. Can you calculate \( S' \cap (E \cup M) \)?
4. How does the total of all participants in \( W \cap (E \cup P) \) compare to other sets?
5. How would you handle a question about the intersection \( (Z \cup S) \cap (E \cup P) \)?

Tip: Always double-check how unions and intersections interact with complements in set theory!

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

Solve the set theory problem where we analyze survey data about people on various diets and supplements. Learn how to calculate (S' ∩ (E ∪ P)) and (W' ∩ (P ∪ M)) using union and intersection of sets. Ideal for college-level discrete mathematics.

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