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

Question

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

Philoid · Accepted Answer

It is given that On Z⁺, define a ∗ b = a^b

a^bϵ Z⁺, so operation * is binary

We know that ab = ba for a, b ϵ Z⁺

⇒ 1 * 2 =1² and 2 * 1 = 2¹ =2

⇒ 1 * 2 ≠ 2 *1, where 1,2 ϵ Z⁺

⇒ The operation * is not commutative.

Also, we get,

(1 * 2) * 3 = 2³ *3 = 8 * 3 = 24

1 * (2 * 3) = 1 * 3² = 1 * 9 = 9

⇒ (1 * 2) * 3 ≠ 1 * (2 * 3), where 1,2,3 ϵ Z⁺

⇒ The operation * is not associative.