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# Let ∗ be a binary operation on the set Q of rational numbers as follows: a ∗ b = (a – b) 2

Q.9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(iv) $a * b = (a-b)^2$

Find which of the binary operations are commutative and which are associative.

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On the set Q ,the operation * is defined as  $a * b = (a-b)^2$ .It is observed that:

For $a,b \in Q$

$a * b = (a-b)^2$

$b* a = (b-a)^2 = \left [ -\left ( a-b \right ) \right ]^{2} = (a-b)^{2}$

$\therefore$       $a*b = b* a$   for $a,b \in Q$

Hence, the * operation is commutative.

It can be observed that

$(1*2)*3 =(1-2)^{2}*3 = 1*3 =(1-3)^{2}= (-2)^{2} =4$

$1*(2*3) =1*(2-3)^{2} = 1*1 =(1-1)^{2} = 0^{2} =0$

$(1*2)*3 \neq 1*(2*3)$  for all $1,2,3 \in Q$

The operation * is not associative.

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