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For each binary operation * defined belo...

For each binary operation * defined below, determine whether * is commutative or associative.
(i) On `Z`, define `a ∗ b = a – b`
(ii) On `Q`, define `a ∗ b = ab + 1`
(iii) On `Q`, define `a ∗ b = (ab)/(2)`
(iv) On `Z^+`, define `a ∗ b = 2^(ab)`
(v) On `Z^+`, define `a ∗ b = a^(b)`
(vi) On `R – {– 1}`, define `a ∗ b = (a)/(b+1)`

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To determine whether each binary operation defined is commutative or associative, we will analyze each operation step by step. ### (i) On `Z`, define `a ∗ b = a – b` **Commutative:** - Check if \( a * b = b * a \). - \( a * b = a - b \) - \( b * a = b - a \) ...
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