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.

Answers (1)

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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