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 $\mathcal{P}(X)$:

+ & \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.