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)