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Binary Operations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsBinary Operations3 Marks
Question
Check the commutativity and associativity of the following binary operations:
$'*'$ on Q defined by a * $b =( a - b )^2$ for $all a , b \in Q$.
✓

Answer

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

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