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

1.	Let A = N x N 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» (a, b) * (c,d) = (a + c, b + d) = (c + a, d + b) = (c, d) * (a, b) hence commutative [(a, b) * (c, d)] * (e, f) = (a + c, b + d) * (e, f) = (a + c + e, b + d + f) = (a, b) * [(c,d) * (e, f)] hence * is associative. Let (e, f) be the identity element of A (e, f) * (a, b) = (a,b) * (c, f) = (a, b) (a + e, b + 0) = (a, b) ⇒ (e = 0, f = 0) ∉ N hence no identity element.

Let A = N x N 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»

(a, b) * (c,d) = (a + c, b + d)

= (c + a, d + b) = (c, d) * (a, b)

hence commutative

[(a, b) * (c, d)] * (e, f)

= (a + c, b + d) * (e, f)

= (a + c + e, b + d + f)

= (a, b) * [(c,d) * (e, f)]

hence * is associative.

Let (e, f) be the identity element of

A (e, f) * (a, b) = (a,b) * (c, f)

= (a, b) (a + e, b + 0) = (a, b)

⇒ (e = 0, f = 0) ∉ N

hence no identity element.