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 –b ϵ Z. so the operation * is binary.

We can see that 1 * 2 = 1-2 = -1 and 2 * 1 = 2-1 = 1.

⇒ 1 *2 ≠ 2 * 1, where 1,2 ϵ Z.

⇒ the operation * is not commutative.

Also, we get,

(1 * 2) * 3 = (1 – 2) * 3 = -1 * 3 = -1 -3 = -4

1 * (2 * 3) = 1 * (2 – 3) = 1 * -1 = 1 – (-1) = 2

⇒ the operation * is not associative.