Let X be a non-empty set and let R be the power set  of X. For a,b \in X, we define the operations a + b := (a \cup b) \setminus (a \cap b) and a \cdot b := a \cap b. Provide a addition table for the set X = {a,b,c}.

Given the set \( X = \{a, b, c\} \) and the operations defined as \( a + b := (a \cup b) \setminus (a \cap b) \) and \( a \cdot b := a \cap b \), we need to construct an addition table for the power set \( \mathcal{P}(X) \).

First, we list the elements of the power set \( \mathcal{P}(X) \):

\[ \mathcal{P}(X) = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\}\} \]

Now, we'll compute the sum \( A + B = (A \cup B) \setminus (A \cap B) \) for each pair of elements \( A, B \in \mathcal{P}(X) \). This operation is essentially the symmetric difference, \( A \Delta B \).

### Addition Table

Let's construct the addition table for the elements of \( \mathcal{P}(X) \):

\[
\begin{array}{c|c|c|c|c|c|c|c|c}
+ & \emptyset & \{a\} & \{b\} & \{c\} & \{a, b\} & \{a, c\} & \{b, c\} & \{a, b, c\} \\
\hline
\emptyset & \emptyset & \{a\} & \{b\} & \{c\} & \{a, b\} & \{a, c\} & \{b, c\} & \{a, b, c\} \\
\{a\} & \{a\} & \emptyset & \{a, b\} & \{a, c\} & \{b\} & \{c\} & \{a, b, c\} & \{b, c\} \\
\{b\} & \{b\} & \{a, b\} & \emptyset & \{b, c\} & \{a\} & \{a, b, c\} & \{c\} & \{a, c\} \\
\{c\} & \{c\} & \{a, c\} & \{b, c\} & \emptyset & \{a, b, c\} & \{a\} & \{b\} & \{a, b\} \\
\{a, b\} & \{a, b\} & \{b\} & \{a\} & \{a, b, c\} & \emptyset & \{b, c\} & \{a, c\} & \{c\} \\
\{a, c\} & \{a, c\} & \{c\} & \{a, b, c\} & \{a\} & \{b, c\} & \emptyset & \{a, b\} & \{b\} \\
\{b, c\} & \{b, c\} & \{a, b, c\} & \{c\} & \{b\} & \{a, c\} & \{a, b\} & \emptyset & \{a\} \\
\{a, b, c\} & \{a, b, c\} & \{b, c\} & \{a, c\} & \{a, b\} & \{c\} & \{b\} & \{a\} & \emptyset \\
\end{array}
\]

This table represents the addition operation \( A + B = (A \cup B) \setminus (A \cap B) \) for all pairs of subsets \( A, B \) of the set \( X \).

Do you want more details or have any questions about this table?

Here are 5 questions you might want to ask:
1. How is the operation \( \cdot \) defined on the power set?
2. Can you provide the multiplication table for the power set of \( X \)?
3. What are the properties of the operation \( + \) in the context of this power set?
4. How would these operations apply if the set \( X \) had more elements?
5. Can you explain the concept of symmetric difference in set theory?

**Tip:** The symmetric difference operation \( A + B \) is equivalent to the set of elements that are in either of the sets \( A \) or \( B \) but not in their intersection.

Explore the addition table for the power set of set X = {a, b, c}, defined with operations based on set theory concepts like symmetric difference. This table illustrates the operation A + B = (A ∪ B) \ (A ∩ B) for all subsets A, B of set X.

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