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 × 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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