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 = a – b

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Answer

For Commutativity: a * b = a - b and b * a = b - a = -(a - b) $\neq\text{a}*\text{b}$

For associativity: a * (b * c) = a * (b - c) = a - (b - c) = (a - b + c)

Also, (a * b) * c = (a - b) * c = (a - b - c)

$\therefore\ \ \ \text{a}*(\text{b}*\text{c})\neq(\text{a}*\text{b})*\text{c}$

Therefore, the operation * is neither commutative nor associative.

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