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Let * be a binary operation on N, defined by a * b = ab for all a, b ∈ N. Show that * is neither commutative nor associative.

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Commutative:

Consider a, b ∈ N where a * b = ab and b * a = ba

Here, aand bare not equal for a, b ∈ N

We know that

a * b ≠ b * a

Hence, * is not commutative.

Associative:

Consider a, b, c ∈ N

(a * b) * c = ab * c = (ab)c = abc

In the same way

a * (b * c) = a * bc = (b)ca

Here, we know that (a * b) * c ≠ a * (b * c)

Hence, * is not associative.

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