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 $a * b = 2^{ab}$ for all $a, b \in N.$

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Answer

Commutative: Let $\text{a, b}\in\text{N},$
Then $a * b = 2^{ab} = 2^{ba} = b * a$
$\therefore a * b = b * a \therefore *$ is commutative on $N.$
Associative: Let $\text{a, b, c}\in\text{N},$ Then $(\text{a}\ ^*\ \text{b}) *\ \text{c}=2^{\text{ab}}\ ^*\ \text{c}=2^{2{\text{ab}}.\text{c}}\ ...(\text{i})$
and, $\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})=\text{a}\ ^*\ 2^{\text{bc}}=2^{\text{a}.2^{\text{bc}}}\ ....(\text{ii})$ From $(i)$ and $(ii),$
we get $(\text{a}\ ^*\ \text{b})\ ^*\ \text{c}\neq\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})$
$\therefore *$ is not associative on $N.$

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