Question

Let $G = P(N)$, where the operation is
\[ A \Delta B = A \cup B - A \cap B \] Which of the following is true?

• A. $G$ is abelian but not cyclic
• B. $G$ has elements of order 4
• C. $G$ has elements of order 8
• D. $\emptyset$ is the identity element of $G$

Hint:

Power set under symmetric difference forms an abelian group. Every element has order 2 and identity is the empty set.

Answer 1

The Correct Option is A

Step 1: Identify the operation.
The operation $A \Delta B = (A \cup B) - (A \cap B)$ is called symmetric difference.
Symmetric difference is known to satisfy:
• Closure
• Associativity
• Commutativity
• Identity element exists
• Every element is its own inverse
Step 2: Identity element.
For any set $A$, \[ A \Delta \emptyset = A \] Hence, $\emptyset$ is the identity element.
Step 3: Check commutativity.
Since $A \Delta B = B \Delta A$, the group is abelian.
Step 4: Order of elements.
For any set $A$, \[ A \Delta A = \emptyset \] So every element has order 2.
Hence, no element can have order 4 or 8.
Step 5: Cyclic property.
Since every non-identity element has order 2, the group cannot be cyclic (except trivial cases).
Thus, $G$ is abelian but not cyclic.
Step 6: Conclusion.
Correct statements are (A) and (D).