For Commutativity,
Let (a, b), (c, d) ∈ R × R
(a, b) * (c, d) = (a + c, b + d)
and (c, d) * (a, b) = (c + a, d + b)
= (a + c, b + d) [∵ Commutative law holds for real number]
⇒ (a, b) * (c, d) = (c, d) * (a, b)
Hence, * is commutative.
For Associativity,
Let (a, b), (c, d) and (e, f) ∈ R × R
((a, b) * (c, d)) * (e, f) = (a + c, b + d) * (e, f) = (a + c + e, b + d + f)
(a, b) * ((c, d) * (e, f)) = (a, b) * (c + e, d + f) = (a + c + e, b + d + f)
((a, b) * (c, d)) * (e, f)) = (a, b) * ((c . d) * (e, f ))
∴ * is associative,
Let (e1,e2) be identity
⇒ (a, b) * (e1, e2) = (a, b)
⇒ (a + e1 , b + e2) = (a, b)
⇒ a + e1 = a and b + e2 = b
⇒ e1 = 0, e2 = 0
(0, 0) ∈ R × R is the identity element.