GATE CS 2023 Q41 — Monoids and Groups

To determine the correct statements about the algebraic structure $H = (2^X, \Delta)$, we verify the group axioms for the symmetric difference operation $\Delta$:

1. Closure: For any $A, B \in 2^X$, their symmetric difference $A \Delta B$ is also a subset of $X$, so $A \Delta B \in 2^X$.
2. Associativity: The symmetric difference 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 $A \in 2^X$, its inverse $A^{-1}$ must satisfy $A \Delta A^{-1} = \emptyset$. Testing $A$ as its own inverse:
   $$A \Delta A = (A - A) \cup (A - A) = \emptyset \cup \emptyset = \emptyset$$Thus, the inverse of $A$ is $A$ itself.

Since all group axioms are satisfied, $H = (2^X, \Delta)$ is a group. This makes Option A correct and Option B incorrect.

Since the inverse of $A$ is $A$ itself, Option D is correct. Option C is incorrect because $A \Delta A^c = X \neq \emptyset$ (for any non-empty set $X$).