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

(v) On Z^+ , define a * b = a^b

Answers (1)

(v) On Z^+ , define a * b = a^b

         1\ast 2 = 1^{2}= 1         and      2\ast 1 = 2^{1}= 2

             \Rightarrow               1\ast 2\neq 2\ast 1       for 1,2 \in Z^{+}

            \therefore  the operation is not commutative.

         (2\ast 3)\ast 4 = 2^{3} \ast 4 = 8 \ast 4 = 8^{ 4}=2^{12}

            2\ast (3\ast 4) = 2 \ast 3^{ 4} = 2 \ast 81 = 2^{81}

              \therefore             (2\ast 3)\ast 4\neq 2\ast (3\ast 4) ;     where 2,3,4 \in Z^{+}

\therefore  operation * is not  associative.   

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