Question

Let * be a binary operation on N, defined by a * b = a^b for all a, b ∈ N. Show that * is neither commutative nor associative.

Answer 1

Commutative:

Consider a, b ∈ N where a * b = a^b and b * a = b^a

Here, a^b and b^a are not equal for a, b ∈ N

We know that

a * b ≠ b * a

Hence, * is not commutative.

Associative:

Consider a, b, c ∈ N

(a * b) * c = a^b * c = (a^b)^c = a^bc

In the same way

a * (b * c) = a * b^c = (b)^ca

Here, we know that (a * b) * c ≠ a * (b * c)

Hence, * is not associative.