Question

For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.
On Q, define a ∗ b = ab/2

Answer 1

Answer:

On Q, define a * b = \frac{ab}{2}

                  ab = ba for all a,b \in Q

                  {ab}/{2}={ba}/{2}      for all a,b \in Q

              a\ast b=b\ast a       for a,b \in Q

\therefore operation * is commutative.

        (a*b)*c = \frac{ab}{2}*c = \frac{(\frac{ab}{2})c}{2} = \frac{abc}{4}

           a*(b*c) = a*\frac{bc}{2} = \frac{a(\frac{bc}{2})}{2} = \frac{abc}{4}

             \therefore             (a*b)*c=a*(b*c) ;    

\therefore  operation * is  associative.