Q. 145.0( 2 Votes )

For all a, b <spa

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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