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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS2 Marks
Question
Let 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{2}$ for all $\text{a},\text{b}\in\text{Q}_0.$
Show that 'o' is both commutative and associate.
✓

Answer

We have,

 $\text{a }^*\text{ b}=\frac{\text{ab}}{2}$ for all $\text{a},\text{b}\in\text{Q}_0$

Commutativity:

Let $\text{a},\text{b}\in\text{Q}_0,$ then

$\Rightarrow\text{a }^*\text{ b}=\frac{\text{ab}}{2}=\frac{\text{ba}}{2}=\text{a }^*\text{ b}$

$\Rightarrow\text{a }^*\text{ b}=\text{b }^*\text{ a}$

Thus, * is commutative on Q0.

Associativity:

Let $\text{a},\text{b},\text{c}\in\text{Q}_0,$ then

$\Rightarrow(\text{a }^*\text{ b})\ ^*\ \text{c}=\frac{\text{ab}}{2}\ ....(1)$

and, $\text{a }^*\ (\text{b }^*\text{ c})=\text{a }^*\ \frac{\text{bc}}{2}=\frac{\text{abc}}{4}\ ....(2)$

From (1) & (2)

$(\text{a }^*\text{ b})\ ^*\ \text{c}=\text{a }^*\ (\text{b }^*\text{ c})$

⇒ * is accosiative on Q0.

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