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 $B$).

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.