For each binary operation * defined below, determine whether * is… - Vidyadip

Question

For each binary operation $*$ defined below, determine whether $*$ is commutative or associative.
On $Z^+,$ define $a * b = 2^{ab}$

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Answer

For commutativity: $\mathrm{a}^* \mathrm{~b}=2^{\mathrm{ab}}$ and $b * \mathrm{a}=2^{\mathrm{ba}}=2^{\mathrm{ab}}=\mathrm{a} * \mathrm{~b}$
For associativity: $a^*(b * c)=a * 2^{b c}=(2)$
$ \text { Also, }\left(a^* b\right) * c=\left(2^{a b}\right) * 2=2^{a b} \times c $
$ \therefore a *(b * c) \neq(a * b) * c$
Therefore, the operation * is commutative but not associative.

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