Answer :

To prove: * is neither commutative nor associative


Let us assume that * is commutative


ab = ba for all a,b N


This is valid only for a = b


For example take a = 1, b = 2


12 = 1 and 21 = 2


So * is not commutative


Let us assume that * is associative


(ab)c for all a,b,c N


for all a,b,c N


This is valid in the following cases:


(i) a = 1


(ii) b = 0


(iii) bc = bc


For example, let a = 2,b = 1,c = 3


abc = 2(1 × 3) = 23 = 8


= 2


So * is not associative


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