https://r2.erweima.ai/mathbot/image/1725501322376qg4ra7sy.jpg

Let's analyze the given problem from the image.

We are given two sets defined by:

- \( P = \left\{ 2x - 1 \mid 3 \leq x < 9 \text{ and } x \text{ is even} \right\} \)
- \( Q = \left\{ \frac{3x+1}{2} \mid P/2 < x < 10 \text{ and } x \text{ is odd} \right\} \)

We need to calculate the number of proper subsets of \( Q \).

### Step 1: Understanding Set \( P \)
\( P = \{ 2x - 1 \mid 3 \leq x < 9 \text{ and } x \text{ is even} \} \)

- When \( x = 4 \), \( 2x - 1 = 7 \)
- When \( x = 6 \), \( 2x - 1 = 11 \)
- When \( x = 8 \), \( 2x - 1 = 15 \)

Thus, the set \( P = \{ 7, 11, 15 \} \).

### Step 2: Understanding Set \( Q \)
\( Q = \left\{ \frac{3x+1}{2} \mid \frac{P}{2} < x < 10 \text{ and } x \text{ is odd} \right\} \)

We need to find the elements of \( Q \) based on the values for odd integers between 1 and 9. Let’s evaluate the function \( \frac{3x+1}{2} \) for \( x \):

- For \( x = 1 \), \( \frac{3(1) + 1}{2} = 2 \)
- For \( x = 3 \), \( \frac{3(3) + 1}{2} = 5 \)
- For \( x = 5 \), \( \frac{3(5) + 1}{2} = 8 \)
- For \( x = 7 \), \( \frac{3(7) + 1}{2} = 11 \)
- For \( x = 9 \), \( \frac{3(9) + 1}{2} = 14 \)

Thus, the set \( Q = \{ 2, 5, 8, 11, 14 \} \).

### Step 3: Proper Subsets of \( Q \)
The number of proper subsets of a set with \( n \) elements is \( 2^n - 1 \). 

Since \( Q \) has 5 elements, the number of proper subsets is:

\[
2^5 - 1 = 32 - 1 = 31
\]

However, the question specifically asks for a **subset** of \( Q \), which might need further context based on the given options. Let's check the result options.

Do you need any clarification on this? Here are five additional questions related to this problem:

1. How do we calculate the number of subsets for a set?
2. What is the difference between a subset and a proper subset?
3. How would this problem change if we added more constraints to set \( P \)?
4. Can you find subsets using different methods?
5. How can set notation be used to simplify understanding problems like this?

**Tip**: A proper subset is any subset of a set that does not include all elements of the original set.

Sean los conjuntos definidos por P = {2x - 1 / 3 ≤ x < 9 ∧ x es par} y Q = { (3x+1)/2 | P/2 < x < 10 ∧ x es impar}. Calcule el número de subconjuntos propios de Q.

This problem involves calculating the number of proper subsets for set Q, which is based on even and odd number constraints. It explores set theory concepts and uses the formula for proper subsets (2^n - 1). Suitable for students in Grades 10-12.

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