discrete math question

Question

Define * on Q by p*q=pq+1. Prove or disprove that * is commutative or associative

Answer 1

The operation * is commutative: for any p and q in Q, we have p*q=pq+1. We know the usual multiplication on Q is commutative, so pq=qp. Now q*p=qp+1=pq+1=p*q, that is, q*p=p*q. So * is commutative.

However, the operation is not associative: let p=2, q=3, and s=4. Then (2*3)*4=((2)(3)+1)*4=7*4=(7)(4)+1=29 while 2*(3*4)=2*((3)(4)+1)=2*13=(2)(13)+1=27. That is (2*3)*4 is not equal to 2*(3*4) and so the operation * is not associative in general.