LetA= 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) ε A.

#### Solution

(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 ε R

* is commulative on A

(ii) Associative : ______

(a, b), (c, d), (e, f) ε 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) ε A

(a + x, b+y) = (a, b) for all (a, b) ε A

a + x = a, b + y = b for all (a, b) ε A

x = 0, y = 0

(0, 0) ε 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)