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