Let $X$ be a set and $2^X$ denote the powerset of $X$. Define a binary operation $\Delta$ on $2^X$ as follows: $$A \Delta B = (A - B) \cup (B - A) \ .$$ Let $H = (2^X, \Delta)$. Which of the following statements about $H$ is/are correct?

To determine whether $H = (2^X, \Delta)$ is a group and find the inverses of its elements, we check the group axioms for the symmetric difference operation $\Delta$ on the power set $2^X$:

1. Closure: For any $A, B \in 2^X$, the symmetric difference $A \Delta B = (A - B) \cup (B - A)$ is a subset of $X$, so $A \Delta B \in 2^X$.

2. Associativity: The symmetric difference of sets is associative, i.e., $(A \Delta B) \Delta C = A \Delta (B \Delta C)$ for all $A, B, C \in 2^X$.

3. Identity Element: The empty set $\emptyset \in 2^X$ acts as the identity element because:

   $$A \Delta \emptyset = (A - \emptyset) \cup (\emptyset - A) = A \cup \emptyset = A$$

4. Inverse Element: For any set $A \in 2^X$:

   $$A \Delta A = (A - A) \cup (A - A) = \emptyset \cup \emptyset = \emptyset$$

   Since $A \Delta A = \emptyset$ (the identity element), every element $A$ is its own inverse.

Thus:

• (A) is correct: $H$ is a group (specifically, a Boolean group).

• (B) is incorrect: $H$ is indeed a group.

• (C) is incorrect: The inverse of $A$ is $A$, not its complement $A^c$ (since $A \Delta A^c = X \neq \emptyset$ in general).

• (D) is correct: For every $A \in 2^X$, the inverse of $A$ is $A$.

Hence, the correct options are A and D.