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RELATIONS AND FUNCTIONS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRELATIONS AND FUNCTIONS3 Marks
Question
Check the commutativity and associativity of the following binary operations : $'*'$ on $Q$ defined by $a * b = (a - b)^2$ for all $a, b \in Q$.
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Answer

Commutativity:
Let $\text{a, b}\in\text{Q}.$ Then$, a * b = (a - b)^2$
$= (b - a)^2$
$= b * a$
Therefore,
$a * b = b * a, \forall\ \text{a, b}\in\text{Q}$
Thus$, *$ is commutative on $Q$.
Associativity:
Let $\text{a, b, c}\in\text{Q}.$ Then,
$a * (b * c) = a * (b - c)^2$
$= a * (b^2 + c^2 - 2bc)$
$= (a - b^2 - c^2 + 2bc)^2$
$(a * b) * c = (a - b)^2 * c$
$= (a^2 + b^2 - 2ab) * c$
$= (a^2 + b^2 - 2ab - c)^2$
Therefore,
$\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})\neq(\text{a}\ ^*\ \text{b})\ ^*\ \text{c}$
Thus$, *$ is not associative on $Q$.

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