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# Let * be a binary operation on the set Q of rational numbers as follows: a * b = ab over 4

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

(v)

$a * b = \frac{ab}{4}$

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  $a * b = \frac{ab}{4}$ .It is observed that:

For $a,b \in Q$

$a * b = \frac{ab}{4}$

$b* a = \frac{ba}{4}$

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

Hence, the * operation is commutative.

It can be observed that

$(a*b)*c =(\frac{ab}{4})*c = \frac{\frac{ab}{4}c}{4}=\frac{abc}{16}$

$a*(b*c) =a*(\frac{bc}{4}) = \frac{\frac{bc}{4}a}{4}=\frac{abc}{16}$

$(a*b)*c = a*(b*c)$  for all $a,b,c \in Q$

The operation * is associative.

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