Question:
On the power set \( P(A) \) of a non-empty set \( A \), define:
\[ X \Delta Y = (X \cap Y) \cup (X \cap Y) \]
Which of the following is true?
- (a) Commutative and associative without an identity
- (b) Commutative but not associative with an identity
- (c) Associative but not commutative without an identity
- (d) Associative and commutative with an identity
Solution:
Step 1: Simplify operation
\[ (X \cap Y) \cup (X \cap Y) = X \cap Y \]
So, the operation becomes:
\[ X \Delta Y = X \cap Y \]
—Step 2: Check Commutativity
\[ X \cap Y = Y \cap X \]
So, operation is commutative.
—Step 3: Check Associativity
\[ (X \cap Y) \cap Z = X \cap (Y \cap Z) \]
So, operation is associative.
—Step 4: Identity Element
We need a set \( E \) such that:
\[ X \cap E = X \]
This happens when \( E = A \) (universal set).
So, identity exists.
—Final Answer:
\[ \boxed{\text{(d) Associative and commutative with an identity}} \]