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Question

Let A=R×R and be a binary operation on A defined by (a,b)(c,d)=(a+c,b+d). Show that on A. Also, find the inverse of every element (a,b)A.

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Solution

Commutativity:
Let (a,b),(c,d)R×R
Now, (a,b)×(c,d)=(a+c,b+d)=(c+a,d+b)
[commutativity holds in for ast]
=(c,d)(a,b)
i.e., is commutative.
Associativity:
Let (a,b),(c,d),(e,f)R×R
Now, {(a,b)(c,d)}(e,f)=(a+c,b+d)(e,f)
=(a+c+e,b+d+f)....(1)
Also, (a,b){(c,d)(e,f)}=(a,b)(c+e,d+f)=(a+c+e,b+d+f)...(2)
From (1) and (2), we have
(a,b)(c,d))(e,f)=(a,b)((c,d)(e,f))
i.e., is associative.
Existence of identity element:
Let (e,f) be identity element for in R×R
(a,b)(e,f)=(a,b)=(e,f)(a,b)
(a+e,b+f)=(a,b)=(e+a,f+b)
a+e=a and b+f=b
e=0,f=0, also (0,0)R×R
Hence, (0,0) is the identity element for on A.
(i) for inverse of (a,b)
Let (x,y) be inverse of (a,b)
(a,b)(x,y)=(0,0)
(a+x,b+y)=(0,0)
On equating, we have
a+x=0;b+y=0
x=a;y=b
Hence, inverse of (a,b) is (a,b)

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