Let's break down the content in the image step by step. The problem is presented in the context of set theory with specific sets and operations.

Definitions of the Sets:

• $S = \{ x \mid 1 \leq x \leq 10, x \in \mathbb{N} \}$

  • This means $S$ is the set of natural numbers between 1 and 10, inclusive: $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$

• $A = \{ x \mid 1 \leq x < 10, x \text{ is an odd number} \}$

  • $A$ is the set of odd natural numbers between 1 and 9: $A = \{1, 3, 5, 7, 9\}$

• $B = \{ x \mid 1 \leq x < 7, x \text{ is prime} \}$

  • $B$ is the set of prime numbers less than 7: $B = \{2, 3, 5\}$

• $C = \{2, 3, 8, 10\}$

  • $C$ is a set explicitly given as: $C = \{2, 3, 8, 10\}$

Problems:

(1) $(A \oplus B) - C$

• $\oplus$ refers to the symmetric difference, meaning: $A \oplus B = (A \cup B) - (A \cap B)$ First, compute $A \cup B$ and $A \cap B$:

  • $A \cup B = \{1, 3, 5, 7, 9, 2\} = \{1, 2, 3, 5, 7, 9\}$
  • $A \cap B = \{3, 5\}$ So, $A \oplus B = (A \cup B) - (A \cap B) = \{1, 2, 7, 9\}$.

  Now, subtract $C$ from this result: $(A \oplus B) - C = \{1, 2, 7, 9\} - \{2, 3, 8, 10\} = \{1, 7, 9\}$

(2) $(B \cup C)^c$

• The complement $(B \cup C)^c$ means the elements in $S$ that are not in $B \cup C$. First, compute $B \cup C$: $B \cup C = \{2, 3, 5, 8, 10\}$ Now, the complement relative to $S$: $(B \cup C)^c = S - (B \cup C) = \{1, 4, 6, 7, 9\}$

(3) $(A \cap C) - (B \oplus C)$

• First, calculate $A \cap C$ and $B \oplus C$:

  • $A \cap C = \{2, 3, 8, 10\} \cap \{1, 3, 5, 7, 9\} = \{3\}$
  • $B \oplus C = (B \cup C) - (B \cap C) = \{2, 3, 5, 8, 10\} - \{2, 3\} = \{5, 8, 10\}$

  Now, subtract $B \oplus C$ from $A \cap C$: $(A \cap C) - (B \oplus C) = \{3\} - \{5, 8, 10\} = \{3\}$

Summary of Results:

1. $(A \oplus B) - C = \{1, 7, 9\}$
2. $(B \cup C)^c = \{1, 4, 6, 7, 9\}$
3. $(A \cap C) - (B \oplus C) = \{3\}$

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

Here are some follow-up questions to expand your understanding:

1. How would the results change if $A$ or $B$ included more elements?
2. Can you explain the difference between union, intersection, and symmetric difference?
3. How would you find the complement of a set if the universal set were larger?
4. What is the significance of prime numbers in the context of set $B$?
5. How would the results change if $C$ had fewer elements?

Tip: When solving set operations, always compute smaller components like union and intersection first to simplify the overall calculation.