Let's break down the problem step by step:

You are given a universal set $S = \{ 1, 2, 3, \dots, 15 \}$, and two subsets:

• $A = \{ x \mid x \text{ is a prime number between 5 and 15} \}$
• $B = \{ x \mid x \text{ is an odd number between 5 and 15} \}$

Step 1: Define the sets $A$ and $B$

Set $A$ (Prime numbers between 5 and 15):

Prime numbers between 5 and 15 are: $5, 7, 11, 13$.

So, $A = \{ 5, 7, 11, 13 \}$.

Set $B$ (Odd numbers between 5 and 15):

Odd numbers between 5 and 15 are: $5, 7, 9, 11, 13, 15$.

So, $B = \{ 5, 7, 9, 11, 13, 15 \}$.

Step 2: Perform the set operations

(a) $A \cup B$ (Union of $A$ and $B$):

The union of two sets includes all elements that are in either $A$ or $B$.

$A \cup B = \{ 5, 7, 9, 11, 13, 15 \}$

(b) $A \cap B$ (Intersection of $A$ and $B$):

The intersection of two sets includes all elements that are in both $A$ and $B$.

$A \cap B = \{ 5, 7, 11, 13 \}$

(c) $B^c$ (Complement of $B$ with respect to $S$):

The complement of $B$ includes all elements in $S$ that are not in $B$.

$B^c = S - B = \{ 1, 2, 3, 4, 6, 8, 10, 12, 14 \}$

(d) $A - B$ (Difference of $A$ and $B$):

The difference of $A$ and $B$ includes all elements that are in $A$ but not in $B$.

$A - B = \emptyset \quad (\text{since all elements of } A \text{ are also in } B)$

Summary of results:

• (a) $A \cup B = \{ 5, 7, 9, 11, 13, 15 \}$
• (b) $A \cap B = \{ 5, 7, 11, 13 \}$
• (c) $B^c = \{ 1, 2, 3, 4, 6, 8, 10, 12, 14 \}$
• (d) $A - B = \emptyset$

Let me know if you'd like further explanation or have any additional questions!

Here are five related questions you can try:

1. What is $A^c$, the complement of $A$?
2. What is $S - A$, the difference of $S$ and $A$?
3. What is $A \cup A^c$ in relation to the universal set $S$?
4. Can a prime number ever be an even number?
5. What is the symmetric difference between sets $A$ and $B$?

Tip: When working with set operations, always ensure you fully understand the definitions of union, intersection, and difference, as they are fundamental to solving problems involving sets.