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Define * on Z by a * b = a + b – ab. Show that * is a binary operation on Z which is commutative as well as associative.

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Consider a, b ∈ Z where a * b = a + b – ab and b * a = b + a – ba

So we get a * b = b * a

Associative:

Consider a, b, c ∈ Z

Here,

(a * b) * c = (a + b – ab) * c = a + b – ab + c – (a + b – ab) c

We get

(a * b) * c = a + b + c – ab – bc – ca + abc

a * (b * c) = a * (b + c – bc) = a + b + c – bc – a (b + c – bc)

We get

a * (b * c) = a + b + c – an – bc – ca + abc

So (a * b) * c = a * (b * c)

Therefore, operation on Z is associative.

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