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Binary Operations — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSBinary Operations3 Marks
Question
Check the commutativity and associativity of the following binary operations:
'o' on Q defined by $\text{a o b}=\frac{\text{ab}}{2}$ for all a, b ∈ Q.
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Answer

Binary operation 'o' defined on Q, given by $\text{a o b}=\frac{\text{ab}}{2}$ for all a, b ∈ Q.
Commutative: Let $\text{a, b}\in\text{Q},$ Then
$\text{a o b}=\frac{\text{ab}}{2}=\frac{\text{ba}}{2}=\text{b o a}$
$\Rightarrow\ \text{a o b}=\text{b o a}$
$\therefore$ o is commutative on Q.
Associativity: Let $\text{a, b, c}\in\text{Q},$ Then,
$(\text{a o b})\text{ o c}=\Big(\frac{\text{ab}}{2}\Big)\text{ o c}=\frac{\text{abc}}{4}\ .....(\text{i})$
$\text{a o }(\text{b o c})=\text{a o }\Big(\frac{\text{bc}}{2}\Big)=\frac{\text{abc}}{4}\ ....(\text{ii})$
From (i) and (ii) we get
(a o b) o c = a o (b o c)
$\therefore$ 'o' is associative on Q.

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