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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
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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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