The symmetric difference between sets, denoted as $A \oplus B$, is defined as the set of elements that are in either $A$ or $B$, but not in both. Mathematically:

$A \oplus B = (A \cup B) \setminus (A \cap B).$

To compute $A \oplus (B \oplus C)$, we proceed step by step:

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Step 1: Compute $B \oplus C$

Given: $B = \{0, 1, 2, 3, 4, 5, 6\}, \quad C = \{4, 5, 6, 7, 8, 9, 10\}.$

1. Union of $B$ and $C$: $B \cup C = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}.$

2. Intersection of $B$ and $C$: $B \cap C = \{4, 5, 6\}.$

3. Symmetric Difference $B \oplus C$: $B \oplus C = (B \cup C) \setminus (B \cap C) = \{0, 1, 2, 3, 7, 8, 9, 10\}.$

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Step 2: Compute $A \oplus (B \oplus C)$

Now, $A = \{0, 2, 4, 6, 8, 10\}$ and $B \oplus C = \{0, 1, 2, 3, 7, 8, 9, 10\}$.

1. Union of $A$ and $B \oplus C$: $A \cup (B \oplus C) = \{0, 1, 2, 3, 4, 6, 7, 8, 9, 10\}.$

2. Intersection of $A$ and $B \oplus C$: $A \cap (B \oplus C) = \{0, 2, 8, 10\}.$

3. Symmetric Difference $A \oplus (B \oplus C)$: $A \oplus (B \oplus C) = (A \cup (B \oplus C)) \setminus (A \cap (B \oplus C)) = \{1, 3, 4, 6, 7, 9\}.$

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Final Answer:

$A \oplus (B \oplus C) = \{1, 3, 4, 6, 7, 9\}.$

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Would you like further details or examples? Here are some related questions to deepen understanding:

1. How is the symmetric difference related to set difference and union?
2. Can symmetric difference be applied to more than two sets?
3. What are the properties of symmetric difference, such as commutativity and associativity?
4. How is $A \oplus B$ different from $A \cup B$?
5. Can symmetric difference be represented using Venn diagrams?

Tip: When working with symmetric differences, always remember to handle the union and intersection carefully—they are the building blocks of this operation!