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