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 element for ∗ on A, if any.

solution:

A = N x N and * is a paired activity characterized on A. (a, b) * (c, d) = (a + c, b + d)

(c, d) * (a, b) = (c + a, d + b) = (a + c, b + d) The activity * is commutative

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

=> ((a, b) * (c, d)) * (e, f) = (a, b) * ((c, d)) * (e, f)) The activity * is acquainted.

Let (e, f) be the character work, then, at that point (a, b) * (e, f) = (a + e, b + f)

For character work, a = a + e => e = 0 and b = b + f => f = 0

As zero isn’t a piece of set of normal numbers. So character work doesn’t exist. As 0 ∉ N, subsequently, personality component doesn’t exist.