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 Q, define a ∗ b = ab + 1

Philoid · Accepted Answer

It is given that On Q, define a ∗ b = ab + 1

ab + 1 ϵ Q, so operation * is binary

We know that ab = ba for a,b ϵ Q

⇒ ab + 1 = ba + 1for a,b ϵ Q

⇒ a * b = a * b for a,b ϵ Q

⇒ 1 *2 ≠ 2 * 1, where 1,2 ϵ Z.

⇒ The operation * is commutative.

Also, we get,

(1 * 2) * 3 = (1 × 2) * 3 = 3 * 3 = 3 × 3 + 1 = 10

1 * (2 * 3) = 1 * (2 × 3) = 1 * 7 = 1 × 7 + 1 = 8

⇒ (1 * 2) * 3 ≠ 1 * (2 * 3)

⇒ the operation * is not associative.