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

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

(ii) $a*b = a^2 + b^2$

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

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On the set Q ,the operation * is defines as  $a*b = a^2 + b^2$.It is observed that:

For $a,b \in Q$

$a*b=a^{2}+b^{2}= b^{2}+a^{2}=b*a$

$\therefore$      $a*b=b*a$

Hence, the * operation is commutative.

It can be observed that

$(1*2)*3 =(1^{2}+2^{2})*3 = 5*3 = 5^{2}+3^{2} = 25+9 =34$

$1*(2*3) =1*(2^{2}+3^{2}) = 1*13 = 1^{2}+13^{2} = 1+169 =170$

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

The operation * is not associative.

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