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

On Z⁺, define a ∗ b = 2^ab

It is given that on Z⁺, define a ∗ b = 2^ab

2^abϵ Z⁺, so operation * is binary

We know that ab = ba for a,b ϵ Z⁺

⇒ 2^ab = 2^ba for a,b ϵ Z⁺

⇒ a * b = a * b for a, b ϵ Z⁺

⇒ The operation * is commutative.

Also, we get,

(1 * 2) * 3 = 2^(1×2) *3 = 4 * 3 = 2^(4×3) = 2¹²

1 * (2 * 3) = 1 * 2^{(2 × 3)} = 1 * 2⁶ = 1 × 64 =2⁶⁴

⇒ (1 * 2) * 3 ≠ 1 * (2 * 3)

⇒ The operation * is not associative.