Q13 of 123 Page 3

The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is

1) Commutative:


⇒ a * b = a + b + ab …(1)


⇒ b * a = b + a + ba …(2)


⇒ a * b= b * a


2) Associative:


⇒ (a * b)* c = (a + b + ab) * c


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


(a + b + ab) * c = a + b + c + ab +ac + bc + abc


⇒ a * (b * c) = a * (b + c + bc)


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


a * (b + c + bc) = a + b + c + ab +ac + bc + abc


⇒ (a * b)* c = a * (b * c)

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