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

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