https://file.aiquickdraw.com/mathbot/file/1727850458667kk8z2zsu.jpg

From the image, we can break down the problem as follows:

Given:

- Set \( S = \{ 1, 2, 3, \ldots, 15 \} \)
- Set \( A \) is defined as \( A = \{ x \mid x \text{ is a prime number between 5 and 15} \} \)
- Set \( B \) is defined as \( B = \{ x \mid x \text{ is an odd number between 5 and 15} \} \)

We are asked to determine the following:

a. \( A \cup B \) (Union of A and B)

b. \( A \cap B \) (Intersection of A and B)

c. \( B^c \) (Complement of B in \( S \))

d. \( A - B \) (Difference of A and B)

### Step 1: Identify the elements of \( A \) and \( B \)

1. **Prime numbers between 5 and 15**:
   The prime numbers are 5, 7, 11, and 13. So, \( A = \{5, 7, 11, 13\} \).

2. **Odd numbers between 5 and 15**:
   The odd numbers are 5, 7, 9, 11, 13, and 15. So, \( B = \{5, 7, 9, 11, 13, 15\} \).

### Step 2: Solve the required operations

a. **Union of \( A \cup B \)**:  
   \( A \cup B = \{5, 7, 9, 11, 13, 15\} \)  
   (Combine all unique elements from sets \( A \) and \( B \)).

b. **Intersection of \( A \cap B \)**:  
   \( A \cap B = \{5, 7, 11, 13\} \)  
   (Elements common to both sets \( A \) and \( B \)).

c. **Complement of \( B \) in \( S \), i.e., \( B^c \)**:  
   \( B^c \) consists of the elements in \( S \) that are not in \( B \).  
   \( B^c = S - B = \{1, 2, 3, 4, 6, 8, 10, 12, 14\} \).

d. **Difference \( A - B \)**:  
   \( A - B \) consists of elements in \( A \) that are not in \( B \).  
   \( A - B = \emptyset \)  
   (Since all elements of \( A \) are also in \( B \)).

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Would you like further details or explanations on any part of this?

### Related Questions:
1. How do we determine the prime numbers between two limits?
2. What is the difference between union and intersection in set theory?
3. How do we calculate the complement of a set within a universal set?
4. What are common operations in set theory, and how are they applied?
5. Can you explain how the set difference operation works with examples?

**Tip:**  
Always list the elements in both sets clearly when performing union, intersection, or difference operations to avoid errors.

Diketahui S = {1, 2, ..., 15}. A = {x | x bilangan prima antara 5 dan 15}, B = {x | x bilangan ganjil antara 5 dan 15}. Tentukan A ∪ B, A ∩ B, B^c, dan A - B.

Learn how to solve a set theory problem involving the union, intersection, complement, and difference of sets. This problem works with prime and odd numbers between 5 and 15. Suitable for students in Grades 7-9.

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