Question

Check the commutativity and associativity of the following binary operations '*'. on N defined by a * b = 2^ab for all a, b ∈ N ?

Answer 1

 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.