# Let <span lang="E

(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,b + d)

a + c = 0 b + d = 0

a = - c b = - d

c = - a d = - b

(a,b) is invertible

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