Show that the binary operation * on defined as for all is commutative and associative on A. Also find the identify element of * in A and prove that every element of A is invertible.

Given: binary operation * on A = R – {-1} defined as

a * b = a + b + ab for all a and b belongs to A

To prove: given operation is is commutative and associative on A and every element of A is invertible.

To find: the identify element of * in A

Commutativity:

a * b = a + b + ab and b * a = b + a + ba

Since, a + b + ab = b + a + ba

⇒ a * b = b * a

This shows that operation is commutative

Associativity:

(a * b) * c

= (a + b + ab) * c

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

= a + b + ab + c + ac + bc + abc……………(1)

and

a * (b * c)

= a * (b + c + bc)

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

= a + b + c + bc + ab + ac + abc………………(2)

Since, (1) = (2)

⇒ (a * b) * c = a * (b * c)

This shows that operation is associative

Existence of identity:

Let e be the identity element

a * e = a = e * a

⇒ a + e + ae = a and e + a + ea = a

⇒ e(1 + a) = a – a

⇒ e(1 + a) = 0

⇒ e = 0

So, 0 is the identity element of the * operation

Existence of inverse:

a * b = e = e * b

⇒ a + b + ab = 0