Question

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

On Z⁺, define a * b = 2^ab

Answer 1

On Z⁺, * is defined by a * b = 2^ab.

It is known that:

ab = ba &mnForE; a, b ∈ Z⁺

⇒ 2^ab = 2^ba &mnForE; a, b ∈ Z⁺

⇒ a * b = b * a &mnForE; a, b ∈ Z⁺

Therefore, the operation * is commutative.

It can be observed that:

`(1*2)*3 = 2^(1xx2) * 3 = 4 * 3 = 2^(4xx3) = 2^12`

`1 * (2 * 3) = 1 * 2^(2 xx 3) = 1 * 2^6  = 1 ** 64 = 2^(64)`

∴(1 * 2) * 3 ≠ 1 * (2 * 3) ; where 1, 2, 3 ∈ Z⁺

Therefore, the operation * is not associative.