Let * be a binary operation on the set Q of rational numbers as f… - Vidyadip

Question

Let * be a binary operation on the set Q of rational numbers as follows:
$\text{a} * \text{b} = \frac{\text{ab}}{4}$

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Answer

$\text{a}*\text{b}=\frac{\text{ab}}{4}=\frac{\text{ba}}{4}=\text{b}*\text{a}$
$\therefore$ operation * is commutative.
$(\text{a}*\text{b})*\text{c}=\frac{\text{ab}}{4}*\text{c}=\frac{\frac{\text{ab}}{4}\text{c}}{4}=\frac{\text{abc}}{16}$
And $\text{a}*(\text{b}*\text{c})=\text{a}*\frac{\text{bc}}{4}=\frac{\text{a}\frac{\text{bc}}{4}}{4}=\frac{\text{abc}}{16}$
Here, $(\text{a}*\text{b})*\text{c}=\text{a}*(\text{b}*\text{c})$
$\therefore$ operation * is associative.

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