Question

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

Answer 1

Commutativity:

\[\text{Let }a, b \in N . \text{Then}, \]

\[a * b = a^b \]

\[b * a = b^a \]

\[\text{Therefore},\]

\[a * b \neq b * a\]

Thus, * is not commutative on N.

Associativity:

\[\text{Let }a, b, c \in N . \text{Then}, \]

\[a * \left( b * c \right) = a * \left( b^c \right)\]

                   \[ = a^{b^c} \]

\[\left( a * b \right) * c = \left( a^b \right) * c\]

                  \[ = \left( a^b \right)^c \]

                  \[ = a^{bc} \]

        \[\text{Therefore},\]

\[a * \left( b * c \right) \neq \left( a * b \right) * c\]

Thus, * is not associative on N.