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प्रश्न
Check the commutativity and associativity of the following binary operations '*'. on N defined by a * b = 2ab for all a, b ∈ N ?
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उत्तर
Commutativity:
\[\text{Let } a, b \in N . \text{Then}, \]
\[a * b = 2^{ab} \]
\[ = 2^{ba} \]
\[ = b * a\]
\[\text{Therefore},\]
\[a * b = b * a, \forall a, b \in N\]
Thus, * is commutative on N.
Associativity:
\[\text{Let a}, b, c \in N . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( 2^{bc} \right)\]
\[ = 2^{a * 2^{bc}} \]
\[\left( a * b \right) * c = \left( 2^{ab} \right) * c\]
\[ = 2^{ab * 2^c} \]
\[\text{Therefore},\]
\[a * \left( b * c \right) \neq \left( a * b \right) * c\]
Thus, * is not associative on N.
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