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

(iii) a * b = a + ab

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 = a + ab .It is observed that:

For a,b \in Q

1 * 2 = 1+1\times 2 =1 + 2 = 3

2 * 1= 2+2\times 1 =2 + 2 = 4

\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+2\times 1})*3 = 3*3 = 3+3\times 3 = 3+9 =12

1*(2*3) =1*(2+3\times 2) = 1*8 = 1+1\times 8 = 1+8 =9

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

The operation * is not associative.

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