Let A = R × R and * be the binary operation on A defined by (a, b) *

1.	Let A = R × R and * be the 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, if any.
Answer» 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.

Let A = R × R and * be the 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, if any.

Answer»

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.