Let * be a binary operation on the setQof rational numbers as follows: (i)a*b=a−b(ii)a*b=a2+b2 (iii)a*b=a+ab(iv)a*b= (a−b)2 (v)(vi)a*b=ab2 Find which of the binary operations are commutative and which are associative.

(i) OnQ, the operation * is defined asa*b = a − b. It can be observed that: and ∴; where Thus, the operation * is not commutative. It can also be observed that: Thus, the operation * is not associative. (ii) OnQ, the operation * is defined asa*b =a2+b2. Fora, b∈Q, we have: ∴a*b = b*a Thus, the operation * is commutative. It can be observed that:(1*2)*3=(12+22)*3=(1+4)*3=5*3=52+32=25+9=34(1*2)*3=(12+22)*3=(1+4)*3=5*3=52+32=25+9=341*(2*3)=1*(22+32)=1*(4+9)=1*13=12+132=1+169=1701*(2*3)=1*(22+32)=1*(4+9)=1*13=12+132=1+169=170∴ (1*2)*3 ≠ 1*(2*3) , where 1, 2, 3 ∈ Q Thus, the operation * is not associative. (iii) OnQ, the operation * is defined asa*b = a + ab. It can be observed that: Thus, the operation * is not commutative. It can also be observed that: Thus, the operation * is not associative. (iv) OnQ, the operation * is defined bya*b= (a − b)2. Fora,b∈Q, we have: a*b= (a − b)2 b*a= (b − a)2= [− (a − b)]2= (a − b)2 ∴a*b = b*a Thus, the operation * is commutative. It can be observed that: Thus, the operation * is not associative. (v) OnQ, the operation * is defined as Fora,b∈Q, we have: ∴a*b=b*a Thus, the operation * is commutative. Fora, b, c∈Q, we have: ∴(a*b) *c=a* (b*c) Thus, the operation * is associative. (vi) OnQ, the operation * is defined asa*b =ab2 It can be observed that: Thus, the operation * is not commutative. It can also be observed that: Thus, the operation * is not associative. Hence, the operations defined in (ii), (iv), (v) are commutative and the operation defined in (v) is associative. read less