For all a, b ∈ N, we define a * b = a 3 + b 3 . Show that * is commutative but not associative.

Question

Philoid · Accepted Answer

let a = 1,b = 2N

a*b = 1³ + 2³ = 9

And b*a = 2³ + 1³ = 9

Hence * is commutative.

Let c = 3

(a*b)*c = 9*c = 9³ + 3³

a*(b*c) = a*(2³ + 3³) = 1*35 = 1³ + 35³

(a*b)*c a*(b*c)

Hence * is not associative.