Check the commutativity and associativity of the following binary… - Vidyadip

Question

Check the commutativity and associativity of the following binary operations:
$'*'$ on $N,$ defined by $a * b = a^b$ for all $a, b \in N.$

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Answer

Commutativity: Let $\text{a, b}\in\text{N}.$ Then,$\text{a}\ ^*\ \text{b}=\text{a}^{\text{b}}\neq\text{b}^{\text{a}}=\text{b}\ ^*\ \text{a}$
$\Rightarrow\ \text{a}\ ^*\ \text{b}\neq\text{b}\ ^*\ \text{a}$
$\Rightarrow '*'$ is not commutative on $N.$
Associativity: Let $\text{a, b, c}\in\text{N}.$ Then,
$(a * b) * c = a^b * c$
$= (a^b)^c = a^{bc} .....(i)$
$\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})=\text{a}\ ^*\ \text{b}^{\text{c}}=(\text{a})^{\text{b}^{\text{c}}}\ ....(\text{ii})$
From $(i)$ and $(ii)$
$\text{a}^{\text{bc}}\neq(\text{a})^{\text{b}^{\text{c}}}$
$(\text{a}\ ^*\ \text{b})\ ^*\ \text{c}\neq\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})$
$\Rightarrow '*'$ is not associative on $N.$

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