Check the commutativity and associativity of the following binary… - Vidyadip

Question

Check the commutativity and associativity of the following binary operations:
'*' on Q defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{4}$ for all a, b ∈ Q.

✓

Answer

Commutativity: Let $\text{a, b}\in\text{Q}.$ Then,$\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{4}$
$=\frac{\text{ba}}{4}$
$=\text{b}\ ^*\ \text{a}$
Therefore,
$\text{a}\ ^*\ \text{b}=\text{b}\ ^*\ \text{a},\ \forall\ \text{a, b}\in\text{Q}$
Thus '*' is commutative on Z.
Associativity: Let $\text{a, b, c}\in\text{Q}.$ Then,
$\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})=\text{a}\ ^*\ \Big(\frac{\text{bc}}{4}\Big)$
$=\frac{\text{a}\big(\frac{\text{bc}}{4}\big)}{4}$
$=\frac{\text{abc}}{16}$
$(\text{a}\ ^*\ \text{b})\ ^*\ \text{c}=\frac{\text{ab}}{4}\ ^*\ \text{c}$
$=\frac{\big(\frac{\text{ab}}{4}\big)\text{c}}{4}$
$=\frac{\text{abc}}{16}$
Therefore,
$\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})=(\text{a}\ ^*\ \text{b})\ ^*\ \text{c},\ \forall\ \text{a, b, c}\in\text{Q}$
Thus, '*' is associative on Q.

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