Determine which of the following binary operations are associativ… - Vidyadip

Question

Determine which of the following binary operations are associative and which are commutative:
* on Q defined by $\text{a}\ ^*\ \text{b}=\frac{\text{a}+\text{b}}{2}$ for all $\text{a, b}\in\text{Q}$

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Answer

$\text{a}\ ^*\ \text{b}=\frac{\text{a}+\text{b}}{2}=\frac{\text{b}+\text{a}}{2}=\text{b}\ ^*\ \text{a,}$
Which shows * is commutative.
Further, $(\text{a}\ ^*\ \text{b})\ ^*\ \text{c}=\Big(\frac{\text{a}+\text{b}}{2}\Big)\ ^*\ \text{c}$
$=\frac{\big(\frac{\text{a}+\text{b}}{2}\big)+\text{c}}{2}=\frac{\text{a}+\text{b}+2\text{c}}{4}$
Further, $\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})=\text{a}\ ^*\ \Big(\frac{\text{b}+\text{c}}{2}\Big)$
$=\frac{\text{a}+\big(\frac{\text{b}+\text{c}}{2}\big)}{2}=\frac{2\text{a}+\text{b}+\text{c}}{2}\neq\frac{\text{a}+\text{b}+2\text{c}}{4}$
Hence, * is not associative.

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