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

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

(i)  $a * b = a - b$

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  $a * b = a - b$.It is observed that:

$\frac{1}{2}*\frac{1}{3}= \frac{1}{2}-\frac{1}{3}=\frac{1}{6}$

$\frac{1}{3}*\frac{1}{2}= \frac{1}{3}-\frac{1}{2}=\frac{-1}{6}$

$\therefore$      $\frac{1}{2}*\frac{1}{3}\neq \frac{1}{3}*\frac{1}{2}$       here $\frac{1}{2},\frac{1}{3} \in Q$

Hence, the * operation is not commutative.

It can be observed that

$(\frac{1}{2}*\frac{1}{3})*\frac{1}{4} = \left ( \frac{1}{2}-\frac{1}{3}\right )*\frac{1}{4}=\frac{1}{6}*\frac{1}{4}=\frac{1}{6}-\frac{1}{4}=\frac{-1}{12}$

$\frac{1}{2}*(\frac{1}{3}*\frac{1}{4})= \frac{1}{2}*\left ( \frac{1}{3} - \frac{1}{4}\right ) = \frac{1}{2}*\frac{1}{12} = \left ( \frac{1}{2} - \frac{1}{12} \right ) = \frac{5}{12}$

$\left ( \frac{1}{2}*\frac{1}{3} \right )*\frac{1}{4}\neq \frac{1}{2}*(\frac{1}{3}*\frac{1}{4})$  for all $\frac{1}{2},\frac{1}{3}, \frac{1}{4} \in Q$

The operation * is not associative.

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