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

(vi) a* b = ab^2

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

Answers (1)

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

For a,b \in Q

1* 2 = 1\times 2^2=1\times 4=4

2* 1 = 2\times 1^2=2\times 1=2

\therefore       1*2\neq 2*1   for 1,2 \in Q 

Hence, the * operation is not  commutative.

It can be observed that

(1*2)*3 = (1\times 2^{2})*3 = 4*3 = 4\times 3^{2}=4\times 9=36

1*(2*3) = 1*(2\times 3^{2}) = 1*18 = 1\times 18^{2}=1\times 324=324

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

The operation * is  not associative.

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