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

Question

Philoid · Accepted Answer

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)