Question

On the power set P of a non-empty set A, we define an operation ∆ by

\[X ∆ Y = \left( \overline{X} \cap Y \right) \cup \left( X \cap \overline{Y} \right)\]

Then which are of the following statements is true about ∆.

Options

• commutative and associative without an identity

• commutative but not associative with an identity

• associative but not commutative without an identity

• associative and commutative with an identity

Answer 1

Associative and commutative with an identity

\[\text{ Commutativity }: \]
\[X ∆ Y = \left( \overline{X} \cap Y \right) \cup \left( X \cap \overline{Y} \right)\]
\[ = \left( \overline{Y} \cap X \right) \cup \left( Y \cap\overline{ X} \right)\]
\[ = Y ∆ X\]
\[\text{ Thus }, \]
\[X ∆ Y = Y ∆ X\]
\[\text{ Hence, ∆ is commutative on A } .\] 

Let \[\phi\] be the identity element for \[∆\] on P.

\[A ∆ \phi = \left( \overline{A} \cap \phi \right) \cup \left( A \cap \overline{\phi} \right)\]
      \[ = \phi \cup A\]
      \[ = A\]
\[\text{ and }, \]
\[\phi ∆ A = \left( \overline{\phi} \cap A \right) \cup \left( \phi \cap \overline{A} \right)\]
      \[ = A \cup \phi\]
      \[ = A\]