To prove: * is neither commutative nor associative

Let us assume that * is commutative

⇒ a^b = b^a for all a,b N

This is valid only for a = b

For example take a = 1, b = 2

1² = 1 and 2¹ = 2

So * is not commutative

Let us assume that * is associative

⇒ (a^b)^c for all a,b,c N

for all a,b,c N

This is valid in the following cases:

(i) a = 1

(ii) b = 0

(iii) bc = b^c

For example, let a = 2,b = 1,c = 3

a^bc = 2^{(1 × 3)} = 2³ = 8

= 2

So * is not associative