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

Q.2 For each operation ∗ defined below, determine whether ∗ is binary, commutative
or associative.

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

Answers (1)

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

                   ab = ba for all a,b \in Q

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

\Rightarrow               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.   

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seema garhwal

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