# 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.

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.

Exams
Articles
Questions