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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.

The problem involves set theory with operations like union, intersection, symmetric difference, and complement. The sets A, B, C, and S are defined, and three specific operations are required: (1) (A ⊕ B) - C, (2) (B ∪ C)ᶜ, (3) (A ∩ C) - (B ⊕ C).

Learn how to perform set theory operations such as symmetric difference, union, intersection, and complement. This problem uses sets A, B, and C, and covers key concepts in set theory. Step-by-step solutions are provided for (A ⊕ B) - C, (B ∪ C)ᶜ, and (A ∩ C) - (B ⊕ C).

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