Let A= R × R and * be a binary operation on A defined by
(a, b) * (c, d) = (a+c, b+d)
Show that * is commutative and associative. Find the identity element for *
on A. Also find the inverse of every element (a, b) ∈ A.

(a, b) * (c, d) = (a + c, b + d)

(i) Commutative
(a, b) * (c, d) = (a+c, b+d)
(c, d) * (a, b) = (c+a, d+b)
for all, a, b, c, d ∈ R
* is commulative on A

(ii) Associative : ______
(a, b), (c, d), (e, f) ∈A
{ (a, b) * (c, d) } * (e, f)
= (a + c, b+d) * (e, f)
= ((a + c) + e, (b + d) + f)
= (a + (c + e), b + (d + f))
= (a*b) * ( c+d, d+f)
= (a*b) {(c, d) * (e, f)}
is associative on A

Let (x, y) be the identity element in A.

then,

(a, b) * (x, y) = (a, b) for all (a,b) ∈ A
(a + x, b+y) = (a, b) for all (a, b) ∈ A
a + x = a, b + y = b for all (a, b) ∈ A
x = 0, y = 0
(0, 0) ∈ A
(0, 0) is the identity element in A.

Let (a, b) be an invertible element of A.
(a, b) * (c, d) = (0, 0) = (c, d) * (a, b)
(a+c, b+d) = (0, 0) = (c+a, d+b)

a + c = 0 b + d = 0
a = - c b = - d
c = - a d = - b

(a, b) is an invertible element of A, in such a case the inverse of (a, b) is (-a, -b).