For all a, b ∈ R, we define a * b = |a – b|. Show that * is commutative but not associative.

Question

Philoid · Accepted Answer

a*b = a - b if a>b

= - (a - b) if b>a

b*a = a - b if a>b

= - (a - b) if b>a

So a*b = b*a

So * is commutative

To show that * is associative we need to show

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

Or ||a - b| - c| = |a - |b - c||

Let us consider c>a>b

Eg a = 1,b = - 1,c = 5

LHS:

|a - b| = |1 + 1| = 2

||a - b| - c| = |2 - 5| = 3

RHS

|b - c| = | - 1 - 5| = 6

|a - |b - c|| = |1 - 6| = | - 5| = 5

As LHS is not equal to RHS * is not associative