Let A = N × N and be the binary operation. On A defined by (a, b) * (

1.	Let A = N × 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 for ‘ * ’ on A if any.
Answer» Given; A = N × N and (a, b) * (c, d) = (a + c, b + d) (c, d) * (a, b) = (c + a, d + b) = (a + c, b + d) = (a, b) * (c, d)) Hence commutative. Now; (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 + c, b + d] * (e, f) = (a + c + e, b + d + f) Hence associative. (a, b) * (e, e) = (a, b) ⇒ (a + e, b + e) = (a, b) ⇒ a + e = a, b + e = b ⇒ e=0, e = 0 ⇒ (0,0) ∉ A So identity element does not exist.

Let A = N × 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 for ‘ * ’ on A if any.

Answer»

Given; A = N × N and

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

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

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

Hence commutative.

Now; (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 + c, b + d] * (e, f)

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

Hence associative.

(a, b) * (e, e) = (a, b) ⇒ (a + e, b + e) = (a, b)

⇒ a + e = a, b + e = b ⇒ e=0, e = 0 ⇒ (0,0) ∉ A

So identity element does not exist.