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Binary Operations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsBinary Operations3 Marks
Question
Let * be a binary operation on $Q_0$ (set of non-zero rational numbers) defined by a * $b=\frac{a b}{5}$ for $a l l a, b \in Q_0$. Show that * is commutative as well as associative. Also, find its identity element if it exists.
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Answer

Commutativity: Let $\text{a, b}\in\text{Q}_0$
$\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{5}$
$=\frac{\text{ba}}{5}$
$=\text{b}\ ^*\ \text{a}$
Therefore, $\text{a}\ ^*\ \text{b}=\text{b}\ ^*\ \text{a},\forall\ \text{a, b}\in\text{Q}_0$
Thus, * is commutative on $Q_0.$​​​​​​​
Associativity: Let $\text{a, b, c}\in\text{Q}_0$
$\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})=\text{a}\ ^*\ \Big(\frac{\text{bc}}{5}\Big)$
$=\frac{\text{a}\big(\frac{\text{bc}}{5}\big)}{5}$
$=\frac{\text{abc}}{25}$
$(\text{a}\ ^*\ \text{b})\ ^*\ \text{c}=\Big(\frac{\text{ab}}{5}\Big)\ ^*\ \text{c}$
$=\frac{\big(\frac{\text{ab}}{5}\big)\text{c}}{5}$
$=\frac{\text{abc}}{25}$
Therefore,
$a^*\left(b^* c\right)=\left(a^* b\right)^* c, \forall a, b, c \in Q_0$
Thus, * is associative on $Q _0$.
Finding identity element:
Let e be the identity element in Z with respect to * such that, $a^* e = a = e ^* a , \forall a \in Q _0$
$a^* e=a$ and $e^* a=a, \forall a \in Q_0$
Implies that $\frac{ ae }{5}= a$ and $\frac{ ea }{5}= a , \forall a \in Q _0$
Implies that $e =5, \forall a \in Q _0[\because a \neq 0]$
Thus, 5 is the identity element in with respect to *.

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