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

(iv) On Z^+ , define a * b = 2^{ab}

Answers (1)

(iv) On Z^+ , define a * b = 2^{ab}

           ab = ba for all a,b \in Z^{+}

                 2ab = 2ba  for all a,b \in Z^{+}

\Rightarrow               a\ast b=b\ast a       for a,b \in Z^{+}

            \therefore  the operation is commutative.

         (1*2)*3 = 2^{1\times 2} * 3 = 4 * 3 = 2^{4\times 3} = 2^{12}

            1*(2*3) = 1 * 2^{2\times 3} = 1 * 64 = 2^{1\times 64}=2^{64}

              \therefore             (1\ast 2)\ast 3\neq 1\ast (2\ast 3) ;     where 1,2,3 \in Z^{+}

\therefore  operation * is not  associative.   

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