For given binary operation $$ defined below, determine whether $$ is binary, commutative or associative.
(iii)On Q, define $a * b = \frac{a b}{2}$

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Solution

On Q, define $a * b = \frac{a b}{2}$
$\frac{a b}{2} \in Q$ , so the operation $$ is binary. It is known that
ab =ba for $a, b \in Q$
Therefore, $\frac{a b}{2} = \frac{b a}{2}$ for $a, b \in Q$
Therefore, $a b = b * a$ for a, $b \in Q$
Therefore, the operation $$ is commutative.
For all $a, b, c \in Q$ , we have
$(a b) * c = (\frac{a b}{2}) * c = \frac{(\frac{a b}{2}) c}{2} = \frac{a b c}{4} a * (b * c) = a * (\frac{b c}{2}) = \frac{a (\frac{b c}{2})}{2} = \frac{a b c}{4}$
Therefore, $(a * b) * c = a * (b * c)$
Therefore, the operation $*$ is associative.

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