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)`

Explanation

To determine whether each binary operation is commutative or associative, we will analyze each operation step by step.

### (i) On `Z`, define `a ∗ b = a – b`

**Commutative:**
To check if the operation is commutative, we need to see if \( a ∗ b = b ∗ a \).

- \( a ∗ b = a - b \)
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