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Question:

For all $a, b \in R$, we define $a^{*} b=|a-b|$.

Show that $*$ is commutative but not associative.

Solution:

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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