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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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