Let A = R × R and * be a binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Show that * is commutative and associative. Find the identity element for * on A. Also find the inverse of every element (a, b) ∈ A.

A = R × R
(a, b) * (c, d) = (a + c, b + d)

Commutative:
Let (a, b), (c, d) ∈ A
(a, b) * (c, d) = (a + c, b + d) = (c + a, d + b) = (c, d) * (a, b) ∀ (a, b), (c, d) ∈ A

• is commutative.

Associative:
Let (a, b), (c, d), (e, f) ∈ 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 + e, d + f) = (a, b) * ((c, d) * (e, f))
∀ (a, b), (c, d), (e, f) ∈ A

• is associative.

Identity element:
Let (e1, e2) ∈ A is the identity element for * operation by definition.
⇒ (a, b) * (e1, e2) = (a, b)
⇒ (a + e1, b + e2) = (a, b)
a + e1 = a, b + e2 = b
⇒ e1 = 0, e2 = 0
⇒ (0, 0) ∈ A
⇒ (0, 0) is the identity element for *.

Inverse:
Let (b1, b2) ∈ A is the inverse of element (a, b) ∈ A then by definition
(a, b) * (b1, b2) = (0, 0)
(a + b1) = 0, b + b2 = 0
⇒ (−a, −b) ∈ A is the inverse of every element (a, b) ∈ A.