We have,
A = R × R and * is a binary operation on A defined by (a, b) * (c, d) = (a + c, b + d).
Now,
(a, b) * (c, d) = (a + c, b + d) = (c + a, d + b)
⇒ (a, b) * (c, d) = (c, d) * (a, b)
So, * is commutative.
Also,
(a, b) * [(c, d) * (e, f)] = (a, b) * (c + e, d + f)
= (a, b) * (c + e, d + f)
= (a + c + e, b + d + f)
= (a + c, b + d) * (e, f)
= [(a, b) * (c, d)] * (e, f)
⇒ (a, b) * [(c, d) * (e, f)] = [(a, b) * (c, d)] * (e, f)
So, * is associative.
Let (x, y) be the binary element for * on A.
(a, b) * (x, y) = (a, b) = (x, y) * (a, b)
⇒ (a + x, b + y) = (a, b)
⇒ a + x = a and b + y = b
⇒ x = 0 and y = 0
Hence, (0, 0) is the binary element for * on A.