Question

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

On Z⁺, define a * b = a^b

Answer 1

On Z⁺, * is defined by a * b = a^b.

It can be observed that:

`1 *2 = 1^2 = 1` and `2 * 1 = 2^1 = 2`

∴ 1 * 2 ≠ 2 * 1 ; where 1, 2 ∈ Z⁺

Therefore, the operation * is not commutative.

It can also be observed that:

`(2 * 3)*4 = 2^3 * 4 = 8 *4 = 8^4 = (2^3)^4 = 2^(12)`

`2 * (3 *4) = 2 * 3^4 = 2 * 81 = 2^81`

∴(2 * 3) * 4 ≠ 2 * (3 * 4) ; where 2, 3, 4 ∈ Z⁺

Therefore, the operation * is not associative.