For all a, b ∈ R, we define a * b = |a – b|. Show that * is commut

1.	For all a, b ∈ R, we define a * b = \|a – b\|. Show that * is commutative but not associative.
Answer» ab = a - b if a>b = - (a - b) if b>a ba = a - b if a>b = - (a - b) if b>a So ab = ba So * is commutative To show that * is associative we need to show (ab)c = a(bc) 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**

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

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

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