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# For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative. On Z + , define a ∗ b = 2^ab

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

(iv) On $Z^+$ , define $a * b = 2^{ab}$

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(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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